Concentrating nonnegative solutions for double phase Choquard problems

Abstract

This paper deals with the multiplicity and concentration phenomenon of nonnegative solutions for the following double phase Choquard equation $$\begin{array}{cc}& -{\mathrm{div}(|\mathrm{\nabla }\mathrm{u}|}^{\mathrm{p}-2}\mathrm{\nabla }\mathrm{u}+{\mathcal{U}}_{\epsilon }{(\mathrm{x})|\mathrm{\nabla }\mathrm{u}|}^{\mathrm{q}-2}\mathrm{\nabla }\mathrm{u})+{\mathrm{V}}_{\epsilon }{(\mathrm{x})(|\mathrm{u}|}^{\mathrm{p}-2}\mathrm{u}+{\mathcal{U}}_{\epsilon }{(\mathrm{x})|\mathrm{u}|}^{\mathrm{q}-2}\mathrm{u})\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,F,(u)\right)f(u)\mathrm{in}{\mathbb{R}}^{\mathrm{N}},\hfill \end{array}$$where $\epsilon$ is a positive parameter, $N\ge 2,$$1<p<q<N,$$q<2p,$$q<p^{\ast }$ with $p^{\ast }=\frac{\mathit{Np}}{N-p},$$0<\mu <p,$ the function $\mathcal{U}:{\mathbb{R}}^N\to \mathbb{R}$ is continuous, ${\mathcal{U}}_{\epsilon }(x)=\mathcal{U}(\epsilon x),$$V:{\mathbb{R}}^N\to \mathbb{R}$ is a continuous potential and satisfies a local minimum condition, $V_{\epsilon }(x)=V(\epsilon x),$$f:\mathbb{R}\to \mathbb{R}$ is a continuous subcritical nonlinearity in the sense of Hardy–Littlewood–Sobolev inequality and F is the primitive of f. Based on the variational methods and topological arguments, the connection between the multiplicity of solutions and the topological structure of the potential at the local minimum points is established.

Introduction

We focus on the multiplicity and concentration phenomenon of nonnegative solutions for the following double phase Choquard problem

1.1

$$\begin{array}{cc}& -{\mathrm{div}(|\mathrm{\nabla }\mathrm{u}|}^{\mathrm{p}-2}\mathrm{\nabla }\mathrm{u}+{\mathcal{U}}_{\epsilon }{(\mathrm{x})|\mathrm{\nabla }\mathrm{u}|}^{\mathrm{q}-2}\mathrm{\nabla }\mathrm{u})+{\mathrm{V}}_{\epsilon }{(\mathrm{x})(|\mathrm{u}|}^{\mathrm{p}-2}\mathrm{u}+{\mathcal{U}}_{\epsilon }{(\mathrm{x})|\mathrm{u}|}^{\mathrm{q}-2}\mathrm{u})\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,F,(u)\right)f(u)\mathrm{in}{\mathbb{R}}^{\mathrm{N}},\hfill \end{array}$$where $\epsilon$ is a positive parameter, $N\ge 2,$$1<p<q<N,$$q<min\{2p,p^{\ast }\}$ with $p^{\ast }=\frac{\mathit{Np}}{N-p},$$0<\mu <p,$ the function $\mathcal{U}:{\mathbb{R}}^N\to \mathbb{R}$ is continuous, ${\mathcal{U}}_{\epsilon }(x)=\mathcal{U}(\epsilon x),$$V:{\mathbb{R}}^N\to \mathbb{R}$ is a continuous potential and satisfies a local minimum condition, $V_{\epsilon }(x)=V(\epsilon x),$$f:\mathbb{R}\to \mathbb{R}$ is a continuous subcritical nonlinearity in the sense of the Hardy–Littlewood–Sobolev inequality and F is the primitive of f.

Suppose that $\mathcal{U}$ and V have the following properties: $(A_1)$

$\mathcal{U}:{\mathbb{R}}^N\to \mathbb{R}$ is a continuous and nonnegative function and $\mathcal{U}\in L^{\infty }({\mathbb{R}}^N);$

$(A_2)$

$V:{\mathbb{R}}^N\to \mathbb{R}$ is continuous and there exists $V_0>0$ fulfilling $V_0:={inf}_{x\in {\mathbb{R}}^N}V(x);$

$(A_3)$

there exists a bounded subset $\mathrm{\Lambda }\subset {\mathbb{R}}^N$ such that $$\begin{array}{c}\hfill V_0=\underset{x\in \mathrm{\Lambda }}{inf}V(x)<\underset{x\in \partial \mathrm{\Lambda }}{min}V(x);\end{array}$$

$(A_4)$

there exists $x_{\min }\in \mathrm{\Lambda }$ such that $V_0=V(x_{\min })$ and $\mathcal{U}(x_{\min })={inf}_{{\mathbb{R}}^N}\mathcal{U}(x):={\mathcal{U}}_0\ge 0$

and that $f:\mathbb{R}\to \mathbb{R}$ is a continuous nonlinearity and satisfies the following assumptions: $(f_1)$

${lim}_{t\to 0}\frac{f(t)}{t^{p-1}}=0;$

$(f_2)$

there exists $r_0\in (q,p_{\mu }^{\ast })$ such that $$\begin{array}{c}\hfill \underset{t\to +\infty }{lim}\frac{f(t)}{t^{r_0-1}}=0,\end{array}$$ where $p_{\mu }^{\ast }=\frac{p(N-\mu )}{2(N-p)};$

$(f_3)$

one has that $$\begin{array}{c}\hfill 0<pF(t):=p{\int }_0^{t}f(\tau )\text{d}\tau \le f(t)t\mathrm{for}\mathrm{any}\mathrm{t}>0;\end{array}$$

$(f_4)$

for any $t\in (0,\infty ),$$\frac{f(t)}{t^{q-1}}$ is increasing.

The principal operator in problem (1.1) is called as the double phase operator$$\begin{array}{c}\hfill -\mathrm{div}(|\mathrm{\nabla }\mathrm{u}{|}^{\mathrm{p}-2}\mathrm{\nabla }\mathrm{u}+\mathcal{U}(\mathrm{x})|\mathrm{\nabla }\mathrm{u}{|}^{\mathrm{q}-2}\mathrm{\nabla }\mathrm{u}),\end{array}$$which arises in the energy functional

1.2

$$\begin{array}{c}\hfill {\int }_{\mathrm{\Omega }}{(|\mathrm{\nabla }u|}^p+{\mathcal{U}(x)|\mathrm{\nabla }u|}^q)\text{d}x.\end{array}$$The functional (1.2) is first proposed by Zhikov [32] in investigating the strongly anisotropic materials. Note that the ellipticity of (1.2) is changed on the set where the weight function equals zero. More precisely, the energy density of (1.2) exhibits ellipticity in the gradient of order p on the points x where $\mathcal{U}(x)$ is positive and of order q on the points x where $\mathcal{U}(x)$ vanishes. From the regularity point of view, the functional (1.2) is of the class of nonuniformly elliptic functionals with nonstandard growth conditions of (p, q)-type, according to Marcellini’s terminology; see [21]. For more details on the double phase problems, we refer the readers to [16, 17, 19, 26].

In the case $p=q$ and $\mathcal{U}\equiv 1,$ Eq. (1.1) becomes the quasilinear Choquard equation

1.3

$$\begin{array}{c}\hfill -{\mathrm{\Delta }}_{p}u+V_{\epsilon }(x){|u|}^{p-2}u={\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,F,(u)\right)f(u)\mathrm{in}{\mathbb{R}}^{\mathrm{N}},\end{array}$$whose existence and concentration of solutions have been studied by several authors. Such as, Alves–Gao–Squassina–Yang [4] obtained the multiplicity and concentrating phenomenon of solutions to Choquard equation (1.3) when $p=2,$f is critical growth and the potential V satisfies a global minimum condition

V

$$\begin{array}{c}\hfill V_{\infty }=\underset{|x|\to \infty }{lim\; inf}V(x)>\underset{x\in {\mathbb{R}}^N}{inf}V(x)=V_0>0,\mathrm{where}{\mathrm{V}}_{\infty }\le \infty .\end{array}$$Under a local minimum condition on V,  Alves–Yang [3] first investigated Eq. (1.3) by using penalization methods. After that, Eq. (1.3) has been widely researched in the literature; see for instance [7, 13, 31].

With respect to (p, q)-Laplacian problems, several interested multiplicity and concentration results of solutions were presented in the last years. In particular, the (p, q)-Laplacian problems are a special form of double phase problem (1.1) in the case that $\mathcal{U}\equiv 1$ and for (1.1) in absence of the Choquard nonlinearity, which is

1.4

$$\begin{array}{c}\hfill -{\mathrm{\Delta }}_{p}u-{\mathrm{\Delta }}_{q}u+V_{\epsilon }(x)u=f(u)\mathrm{in}{\mathbb{R}}^{\mathrm{N}}.\end{array}$$With the global minimum assumption (V),  Ambrosio–Repovš [6] considered the multiplicity and concentration of solutions for (1.4). In the literature [8], the existence and concentration of solution for the problem were derived by Ambrosio by means of variational methods when f satisfies Berestycki–Lions type assumption. Zhang–Zhang–Rǎdulescu [28] dealt with that the nonlinearity is Choquard type. This work can be regard as an extend of [6] and they established similar results. Zuo–Zhang–Rǎdulescu [33] considered the interaction between the Kirchhoff term and the Choquard term in (1.4). For more recent contributions on the topic, we recommend the readers to refer [10, 27, 30].

The study of double phase problem in ${\mathbb{R}}^N$ is currently in the early stages and there are a small number of research results; see [14, 17, 18, 22, 29]. Especially, Zhang–Zuo–Rǎdulescu [29] dealt with the double problem (1.1) with nonlinear terms in polynomial form

1.5

$$\begin{array}{cc}& -{\mathrm{div}(|\mathrm{\nabla }\mathrm{u}|}^{\mathrm{p}-2}\mathrm{\nabla }\mathrm{u}+{\mathcal{U}}_{\epsilon }{(\mathrm{x})|\mathrm{\nabla }\mathrm{u}|}^{\mathrm{q}-2}\mathrm{\nabla }\mathrm{u})+{\mathrm{V}}_{\epsilon }{(\mathrm{x})(|\mathrm{u}|}^{\mathrm{p}-2}\mathrm{u}+{\mathcal{U}}_{\epsilon }{(\mathrm{x})|\mathrm{u}|}^{\mathrm{q}-2}\mathrm{u})=\mathrm{f}(\mathrm{u})\hfill \\ \hfill & \mathrm{in}{\mathbb{R}}^{\mathrm{N}},\hfill \end{array}$$in which we obtained the multiplicity and concentration of solutions.

Inspired by the above works, we for the first time analyze the multiplicity and concentration phenomenon of solutions for (1.1) by means of variational methods and Lusternik–Schnirelmann category theory. The main results are as follows.

Theorem 1.1

Let $(A_1)$–$(A_4)$ and $(f_1)$–$(f_4)$ hold. Suppose that $\delta >0$ fulfills $M_{\delta }\subset {\mathrm{\Lambda }}_{\epsilon }.$ Then there exists ${\epsilon }_{\delta }>0$ such that,  for any $\epsilon \in (0,{\epsilon }_{\delta }),$ Eq. (1.1) admits at least ${\mathrm{cat}}_{{\mathrm{M}}_{\delta }}(\mathrm{M})$ nonnegative solutions. If $u_{\epsilon }$ denotes one of these solutions and ${\eta }_{\epsilon }$ is a global maximum point of $u_{\epsilon },$ one has$$\begin{array}{c}\hfill \underset{\epsilon \to 0}{lim}V(\epsilon {\eta }_{\epsilon })=V_0.\end{array}$$

In order to accomplish the proof of Theorem 1.1, we must overcome the following difficulties. First, we adopt a penalization approach proposed by del Pino–Felmer [12], in which they modified nonlinearity outside of set $\mathrm{\Lambda }.$ But this modification depends on the form of the principle operator. Since the behavior of the double phase operator is a p-Laplacian operator on the set $\{x\in {\mathbb{R}}^N:\mathcal{U}(0)=0\}$ and likes a (p, q)-Laplacian operator on the set $\{x\in {\mathbb{R}}^N:\mathcal{U}(0)>0\},$ it has brought many difficulties to us in using del Pino–Felmer approaches to prove that the modified equations is the solutions to the original equation. Then, note that $\frac{1}{{|x|}^{\mu }}\ast F(u)$ may be unbounded even if u belongs to the corresponding variational space. Thus, some new analytical techniques are used to overcome the effects of the convolution term. Last but not least, because f is meanly continuous, the standard Nehari manifold no longer applies and the generalized Nehari manifold method proposed by Szulkin and Weth [23] is exploited.

The structure of this paper is as follows. In Sect. 2, we give the Musielak–Orlicz–Sobolev spaces and Hardy–Littlewood–Sobolev inequality. In Sect. 3, we construct a new equation and consider the compactness of Palais–Smale sequence for the corresponding variational functional. Section 4 is devoted to the study of a scalar equation. Section 5 gives the multiplicity of solutions for the modified equation. Section 6 proves that the solutions for modified equation are (1.1) if $\epsilon$ is small enough.

Preliminaries

The working spaces, referred to as Musielak–Orlicz–Sobolev spaces, will be introduced in this section. Several fundamental properties of these spaces are presented and discussed.

Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$ be a domain. Recall that for each $r\in [1,\infty ],$ the Lebesgue space $L^r(\mathrm{\Omega })$ is equipped with the norm$$\begin{array}{c}\hfill {\Vert u\Vert }_{L^r(\mathrm{\Omega })}:=\left\{\begin{array}{cc}({\int }_{\mathrm{\Omega }}{|u|}^r{\text{d}x)}^{\frac{1}{r}}\hfill & \mathrm{if}\mathrm{r}\in [1,\infty ),\hfill \\ {\mathrm{esssup}}_{\mathrm{x}\in \mathrm{\Omega }}|\mathrm{u}(\mathrm{x})|\hfill & \mathrm{if}\mathrm{r}=\infty .\hfill \end{array}\right)\end{array}$$When $\mathrm{\Omega }={\mathbb{R}}^N,$ we use the norm ${|u|}_r$ replacing the norm ${\Vert u\Vert }_{L^r(\mathrm{\Omega })}$ for brevity.

For $r\in (1,\infty ),$ let ${\mathcal{D}}^{1,r}({\mathbb{R}}^N)$ express the closure of $C_c^{\infty }({\mathbb{R}}^N)$ with the norm$$\begin{array}{c}\hfill {|\mathrm{\nabla }u|}_r:=({\int }_{{\mathbb{R}}^N}{|\mathrm{\nabla }u|}^r\text{d}x{)}^{\frac{1}{r}}.\end{array}$$We give the Sobolev space $W^{1,r}({\mathbb{R}}^N)$ defined as$$\begin{array}{c}\hfill W^{1,r}({\mathbb{R}}^N)=\left\{u,:,{\mathbb{R}}^N,\to ,\mathbb{R},,\mathrm{measurable},:,{\int }_{{\mathbb{R}}^{\mathrm{N}}},{(|\mathrm{u}|}^{\mathrm{r}},+,{|\mathrm{\nabla }\mathrm{u}|}^{\mathrm{r}},)\text{d}\mathrm{x}<\infty \right\},\end{array}$$endowed with the norm$$\begin{array}{c}\hfill {\Vert u\Vert }_{1,r}={(|u|}_r^r+{|\mathrm{\nabla }u|}_r^r{)}^{\frac{1}{r}}.\end{array}$$It is well known that the Sobolev space $W^{1,r}({\mathbb{R}}^N)$ admits embedding properties.

Lemma 2.1

[1]. If $1<r<N,$ then for all $u\in {\mathcal{D}}^{1,r}(R^N)$ there exists $S_{\ast }>0$ independent on u such that$$\begin{array}{c}\hfill {|u|}_{r^{\ast }}^r\le S_{\ast }^{-1}{|\mathrm{\nabla }u|}_r^r,\end{array}$$where $r^{\ast }=\frac{\mathit{Nr}}{N-r}.$ Moreover,  if $r\in (1,N),$ then $W^{1,r}({\mathbb{R}}^N)$ is continuous embedded in $L^t({\mathbb{R}}^N)$ for any $t\in [r,r^{\ast }]$ and compactly embedded in $L_{\text{loc}}^t({\mathbb{R}}^N)$ for any $t\in [r,r^{\ast }).$

We recall a Lions type conclusion.

Lemma 2.2

[2]. Let $1<r<N.$ If $\{u_n\}$ is a bounded sequence in $W^{1,r}({\mathbb{R}}^N)$ and if

2.1

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\underset{y\in {\mathbb{R}}^N}{sup}{\int }_{B_R(y)}{|u_n|}^r\text{d}x=0,\end{array}$$for some $R>0,$ then $u_n\to 0$ in $L^t({\mathbb{R}}^N)$ for all $t\in (r,r^{\ast }).$

In what follows we introduce the Musielak–Orlicz–Sobolev spaces. Given the function ${\mathcal{H}}_{{\mathcal{U}}_{\epsilon }}:{\mathbb{R}}^N\times [0,\infty )\to [0,\infty )$ as$$\begin{array}{c}\hfill {\mathcal{H}}_{{\mathcal{U}}_{\epsilon }}(x,t)=t^p+{\mathcal{U}}_{\epsilon }(x)t^q,\end{array}$$we define the Musielak–Orlicz Lebesgue space $L^{{\mathcal{H}}_{{\mathcal{U}}_{\epsilon }}}({\mathbb{R}}^N)$ by$$\begin{array}{c}\hfill L^{{\mathcal{H}}_{{\mathcal{U}}_{\epsilon }}}({\mathbb{R}}^N)=\{u:{\mathbb{R}}^N\to \mathbb{R}\mathrm{measurable}:{\int }_{{\mathbb{R}}^{\mathrm{N}}}{\mathcal{H}}_{{\mathcal{U}}_{\epsilon }}(\mathrm{x},|\mathrm{u}|)\text{d}\mathrm{x}<\infty \},\end{array}$$which is endowed with the Luxemburg norm$$\begin{array}{c}\hfill {\Vert u\Vert }_{{\mathcal{H}}_{{\mathcal{U}}_{\epsilon }}}=inf\{\lambda >0:{\int }_{{\mathbb{R}}^N}{\mathcal{H}}_{{\mathcal{U}}_{\epsilon }}\left(x,|,\frac{u}{\lambda },|\right)\text{d}x\le 1\}.\end{array}$$We also provide the function ${\mathcal{H}}_{{\mathcal{U}}_{\epsilon },V_{\epsilon }}:{\mathbb{R}}^N\times [0,\infty )\to [0,\infty )$ by$$\begin{array}{c}\hfill {\mathcal{H}}_{{\mathcal{U}}_{\epsilon },V_{\epsilon }}(x,t)=V_{\epsilon }(x)(t^p+{\mathcal{U}}_{\epsilon }(x)t^q).\end{array}$$Furthermore, we define the Musielak–Orlicz Lebesgue space $L^{{\mathcal{H}}_{{\mathcal{U}}_{\epsilon },V_{\epsilon }}}({\mathbb{R}}^N)$ as$$\begin{array}{c}\hfill L^{{\mathcal{H}}_{{\mathcal{U}}_{\epsilon },V_{\epsilon }}}({\mathbb{R}}^N)=\{u:{\mathbb{R}}^N\to \mathbb{R}\mathrm{measurable}:{\int }_{{\mathbb{R}}^{\mathrm{N}}}{\mathcal{H}}_{{\mathcal{U}}_{\epsilon },{\mathrm{V}}_{\epsilon }}(\mathrm{x},|\mathrm{u}|)\text{d}\mathrm{x}<\infty \},\end{array}$$with the Luxemburg norm$$\begin{array}{c}\hfill {\Vert u\Vert }_{{\mathcal{H}}_{{\mathcal{U}}_{\epsilon },V_{\epsilon }}}=inf\left\{\lambda ,>,0,:,{\int }_{{\mathbb{R}}^N},{\mathcal{H}}_{{\mathcal{U}}_{\epsilon },V_{\epsilon }},\left(x,|,\frac{u}{\lambda },|\right),\text{d},x,\le ,1\right\}.\end{array}$$By the above preparations, we can define the weighted Musielak–Orlicz–Sobolev space ${\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}$ as$$\begin{array}{c}\hfill {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}=\{u:{\mathbb{R}}^N\to \mathbb{R}\mathrm{measurable}:\mathrm{u}\in {\mathrm{L}}^{{\mathcal{H}}_{{\mathcal{U}}_{\epsilon },{\mathrm{V}}_{\epsilon }}}({\mathbb{R}}^{\mathrm{N}})\mathrm{and}|\mathrm{\nabla }\mathrm{u}|\in {\mathrm{L}}^{{\mathcal{H}}_{{\mathcal{U}}_{\epsilon }}}({\mathbb{R}}^{\mathrm{N}})\},\end{array}$$with the norm$$\begin{array}{c}\hfill {\Vert u\Vert }_{\epsilon }={\Vert |\mathrm{\nabla }u|\Vert }_{{\mathcal{H}}_{{\mathcal{U}}_{\epsilon }}}+{\Vert u\Vert }_{{\mathcal{H}}_{{\mathcal{U}}_{\epsilon },V_{\epsilon }}}.\end{array}$$Clearly, ${\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}$ is a separable and reflexive Banach space [17, 22] and the following embedding result holds.

Lemma 2.3

[17, 29]. ${\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}$ is continuously embedded in $W^{1,p}({\mathbb{R}}^N).$ Hence, ${\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}$ is continuously embedded in $L^s({\mathbb{R}}^N)$ for any $s\in [p,p^{\ast }]$ and compactly embedded in $L_{\mathrm{loc}}^s({\mathbb{R}}^N)$ for any $s\in [1,p^{\ast }).$

Take the modular ${\varrho }_{\epsilon }:{\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}\to \mathbb{R}$ as$$\begin{array}{c}\hfill {\varrho }_{\epsilon }(u)={\Vert u\Vert }_{p,\epsilon }^p+{\Vert u\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q.\end{array}$$We have the following useful relationship.

Lemma 2.4

[9, 16]. Let $(A_1)$ and $(A_2)$ hold. Then one has that : 

1. if $u\ne 0,$ then ${\Vert u\Vert }_{\epsilon }=\lambda$ if and only if ${\varrho }_{\epsilon }(\frac{u}{\lambda })=1;$

2. ${\Vert u\Vert }_{\epsilon }<1$ (resp. $>1,$$=1)$ if and only if ${\varrho }_{\epsilon }(u)<1$ (resp. $>1,$$=1);$

3. if ${\Vert u\Vert }_{\epsilon }<1,$ then ${\Vert u\Vert }_{\epsilon }^q\le {\varrho }_{\epsilon }(u)\le {\Vert u\Vert }_{\epsilon }^p;$

4. if ${\Vert u\Vert }_{\epsilon }>1,$ then ${\Vert u\Vert }_{\epsilon }^p\le {\varrho }_{\epsilon }(u)\le {\Vert u\Vert }_{\epsilon }^q;$

5. ${\Vert u\Vert }_{\epsilon }\to 0$ if and only if ${\varrho }_{\epsilon }(u)\to 0;$

6. ${\Vert u\Vert }_{\epsilon }\to \infty$ if and only if ${\varrho }_{\epsilon }(u)\to \infty .$

The Hardy–Littlewood–Sobolev inequality is stated as follows.

Lemma 2.5

[20]. Let $t,r>1,$$0<\mu <N$ with $\frac{1}{t}+\frac{\mu }{N}+\frac{1}{r}=2,$$f\in L^t({\mathbb{R}}^N)$ and $h\in L^r({\mathbb{R}}^N).$ There is a sharp constant $C(t,\mu ,r),$ independent of f and h,  such that$$\begin{array}{c}\hfill {\iint }_{{\mathbb{R}}^{2N}}\frac{f(x)h(x)}{{|x-y|}^{\mu }}\text{d}x\text{d}y\le {C(t,\mu ,r,N)|f|}_t{|h|}_r.\end{array}$$

Modified problem

We shall apply the penalization approaches proposed by del Pino–Felmer [12] to study a modified problem.

Let $a>0$ and $k>k_0,$ where $k_0$ is given in Lemma 3.2. We construct the function$$\begin{array}{c}\hfill \widehat{f}(t):=\left\{\begin{array}{cc}f(t)\hfill & \mathrm{for}|\mathrm{t}|<\mathrm{a},\hfill \\ \frac{V_0}{k}{|t|}^{p-2}t\hfill & \mathrm{for}|\mathrm{t}|\ge \mathrm{a}.\hfill \end{array}\right)\end{array}$$We use ${\chi }_{\mathrm{\Lambda }}$ to denote the characteristic function given by$$\begin{array}{c}\hfill {\chi }_{\mathrm{\Lambda }}(x):=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\mathrm{x}\in \mathrm{\Lambda },\hfill \\ 0\hfill & \mathrm{if}\mathrm{x}\notin \mathrm{\Lambda }.\hfill \end{array}\right)\end{array}$$Let $g:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ be Carathéodory function for each $(x,t)\in {\mathbb{R}}^N\times \mathbb{R}$ denoted as$$\begin{array}{c}\hfill g(x,t):={\chi }_{\mathrm{\Lambda }}(x)f(t)+(1-{\chi }_{\mathrm{\Lambda }}(x))\widehat{f}(t).\end{array}$$Take $G(x,t)={\int }_0^{t}g(x,\tau )\text{d}\tau .$ Putting together $(f_1)$–$(f_4),$ we have the following results. $(g_1)$

uniformly for $x\in {\mathbb{R}}^N,$${lim}_{t\to 0}\frac{g(x,t)}{t^{p-1}}=0;$

$(g_2)$

for any $x\in {\mathbb{R}}^N$ and $t>0,$$g(x,t)t\le f(t)t;$

$(g_3)$

(i) for any $x\in \mathrm{\Lambda }$ and $t>0,$$0<qG(x,t)\le g(x,t)t,$

(ii) for any $x\in {\mathrm{\Lambda }}^c$ and $t\in \mathbb{R},$$0\le pG(x,t)\le g(x,t)t\le \frac{V_0}{k}{t}^p;$

$(g_4)$

(i) for any $x\in \mathrm{\Lambda },$$\frac{g(x,t)}{t^{q-1}}$ is increasing in $(0,\infty ),$

(ii) for any $x\in {\mathrm{\Lambda }}^c,$$t\to \frac{g(x,t)}{{|t|}^{p-1}}$ is increasing in $t\in (0,a).$

We deal with the modified problem

3.1

$$\begin{array}{cc}& -{\mathrm{div}(|\mathrm{\nabla }\mathrm{u}|}^{\mathrm{p}-2}\mathrm{\nabla }\mathrm{u}+{\mathcal{U}}_{\epsilon }{(\mathrm{x})|\mathrm{\nabla }\mathrm{u}|}^{\mathrm{q}-2}\mathrm{\nabla }\mathrm{u})+{\mathrm{V}}_{\epsilon }{(\mathrm{x})(|\mathrm{u}|}^{\mathrm{p}-2}\mathrm{u}+{\mathcal{U}}_{\epsilon }{(\mathrm{x})|\mathrm{u}|}^{\mathrm{q}-2}\mathrm{u})\hfill \\ \hfill & =\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,u)\right)g_{\epsilon }(x,u)\mathrm{in}{\mathbb{R}}^{\mathrm{N}},\hfill \end{array}$$where $g_{\epsilon }(x,u)=g(\epsilon x,u)$ and $G_{\epsilon }(x,u)=G(\epsilon x,u).$ We observe that if u is a solution of problem (3.1) and satisfies$$\begin{array}{c}\hfill u(x)<a\mathrm{for}\mathrm{each}\mathrm{x}\in {\mathrm{\Lambda }}_{\epsilon }^{\mathrm{c}},\end{array}$$then u is a solution of the primary problem (1.1). The variational functional of (3.1) is given as$$\begin{array}{c}\hfill {\mathcal{L}}_{\epsilon }(u)=\frac{1}{p}{\Vert u\Vert }_{p,\epsilon }^p+\frac{1}{q}{\Vert u\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q-\frac{1}{2}{\iint }_{{\mathbb{R}}^{2N}}\frac{G_{\epsilon }(x,u)G_{\epsilon }(y,u)}{{|x-y|}^{\mu }}\text{d}x\text{d}y.\end{array}$$By the Hardy–Littlewood–Sobolev inequality, it is easy to conclude that ${\mathcal{L}}_{\epsilon }$ is differentiable, and its derivative can be expressed as$$\begin{array}{cc}\hfill ⟨{\mathcal{L}}_{\epsilon }^{\prime }(u),v⟩& ={\int }_{{\mathbb{R}}^N}{(|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u\mathrm{\nabla }v+{\mathcal{U}}_{\epsilon }{(x)|\mathrm{\nabla }u|}^{q-2}\mathrm{\nabla }u\mathrm{\nabla }v+V_{\epsilon }{(x)(|u|}^{p-2}uv\hfill \\ \hfill & +{\mathcal{U}}_{\epsilon }{(x)|u|}^{q-2}uv))\text{d}x\hfill \\ \hfill & -{\iint }_{{\mathbb{R}}^{2N}}\frac{G_{\epsilon }(x,u)g_{\epsilon }(x,u)v}{{|x-y|}^{\mu }}\text{d}x\text{d}y\mathrm{for}\mathrm{any}\mathrm{u},\mathrm{v}\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}.\hfill \end{array}$$Recall that u is called as a solution of problem (3.1) if $u\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}$ and $⟨{\mathcal{L}}_{\epsilon }^{\prime }(u),v⟩=0$ for any $v\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}.$

Now, we establish the mountain pass geometry to ${\mathcal{L}}_{\epsilon }.$

Lemma 3.1

${\mathcal{L}}_{\epsilon }$ satisfies the following mountain pass geometry.

1. there exist $c_0,$$\zeta >0$ such that ${\mathcal{L}}_{\epsilon }(u)\ge c_0$ for any $u\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}$ with ${\Vert u\Vert }_{\epsilon }=\zeta ;$

2. there exists $e\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}$ with ${\Vert e\Vert }_{\epsilon }>\zeta$ such that ${\mathcal{L}}_{\epsilon }(e)<0.$

Proof

(i) By $(f_1)$ and $(f_2),$ we infer that for any $(x,t)\in {\mathbb{R}}^N\times \mathbb{R},$$$\begin{array}{c}\hfill |G_{\epsilon }(x,t)|\le C_1{(|t|}^p+{|t|}^{r_0}).\end{array}$$Using this, the Hardy–Littlewood–Sobolev inequality and Lemma 2.3, one deduces that

3.2

$$\begin{array}{cc}& |{\iint }_{{\mathbb{R}}^{2N}}\frac{G_{\epsilon }(x,u)G_{\epsilon }(y,u)}{{|x-y|}^{\mu }}\text{d}x\text{d}y|\le C_2|G_{\epsilon }{(x,u)|}_{\frac{2N}{2N-\mu }}^2\le C_3{(|u|}^p+{|u|}^r{)|}_{\frac{2N}{2N-\mu }}^2\hfill \\ \hfill & \le C_3{(||u|}^p{|}_{\frac{2N}{2N-\mu }}^2{+|u|}^r{|}_{\frac{2N}{2N-\mu }}^2)=C_3{(|u|}_{\frac{2Np}{2N-\mu }}^{2p}{+|u|}_{\frac{2Nr}{2N-\mu }}^{2r})\le C_4{(\Vert u\Vert }_{\epsilon }^{2p}+{\Vert u\Vert }_{\epsilon }^{2r}).\hfill \end{array}$$According to (3.2) and Lemma 2.4, we derive that for any $u\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}$ and ${\Vert u\Vert }_{\epsilon }\le 1,$$$\begin{array}{cc}\hfill {\mathcal{L}}_{\epsilon }(u)& \ge \frac{1}{p}{\Vert u\Vert }_{p,\epsilon }^p+\frac{1}{q}{\Vert u\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q-\frac{C_4}{2}{(\Vert u\Vert }_{\epsilon }^{2p}+{\Vert u\Vert }_{\epsilon }^{2r_0})\hfill \\ \hfill & \ge \frac{1}{q}{(\Vert u\Vert }_{p,\epsilon }^q+{\Vert u\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q)-\frac{C_4}{2}{(\Vert u\Vert }_{\epsilon }^{2p}+{\Vert u\Vert }_{\epsilon }^{2r_0})\hfill \\ \hfill & \ge \frac{1}{2^{q-1}q}{(\Vert u\Vert }_{p,\epsilon }+{\Vert u\Vert }_{q,{\mathcal{U}}_{\epsilon }}{)}^q-C_5{(\Vert u\Vert }_{\epsilon }^{2p}+{\Vert u\Vert }_{\epsilon }^{2r_0})\hfill \\ \hfill & =\frac{1}{2^{q-1}q}{\Vert u\Vert }_{\epsilon }^q-C_6{\Vert u\Vert }_{\epsilon }^{2p}.\hfill \end{array}$$Noting that $q<2p,$ then there exist $c_0$ and $\zeta >0$ with $\zeta <1$ such that ${\mathcal{L}}_{\epsilon }(u)\ge c_0$ for any $u\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}$ with ${\Vert u\Vert }_{\epsilon }=\zeta .$

(ii) Let $u_0\in C_c^{\infty }({\mathbb{R}}^N)$ satisfy $\text{supp}(u_0)\subset {\mathrm{\Lambda }}_{\epsilon },$$u_0\ge 0$ in ${\mathbb{R}}^N$ and $u_0\ne 0.$ We consider the function$$\begin{array}{c}\hfill \beta (t)={\iint }_{{\mathbb{R}}^{2N}}\frac{G_{\epsilon }(x,t{u}_0)G_{\epsilon }(y,t{u}_0)}{{|x-y|}^{\mu }}\text{d}x\text{d}y={\iint }_{{\mathbb{R}}^{2N}}\frac{F(t{u}_0(x))F(t{u}_0(y))}{{|x-y|}^{\mu }}\text{d}x\text{d}y.\end{array}$$$(f_3)$ leads to that

3.3

$$\begin{array}{cc}\hfill {\beta }^{\prime }(t)& =2{\iint }_{{\mathbb{R}}^{2N}}\frac{F(t{u}_0(x))f(t{u}_0(y))u_0(y)}{{|x-y|}^{\mu }}\text{d}x\text{d}y\hfill \\ \hfill & =\frac{2}{t}{\iint }_{{\mathbb{R}}^{2N}}\frac{F(t{u}_0(x))f(t{u}_0(y))t{u}_0(y)}{{|x-y|}^{\mu }}\text{d}x\text{d}y\hfill \\ \hfill & \ge \frac{2}{t}{\iint }_{{\mathbb{R}}^{2N}}\frac{F(t{u}_0(x))\theta F(t{u}_0(y))}{{|x-y|}^{\mu }}\text{d}x\text{d}y\hfill \\ \hfill & \ge \frac{2\theta }{t}\beta (t).\hfill \end{array}$$Integrating (3.3) on [1, t] with $t>1,$ there holds$$\begin{array}{c}\hfill \beta (t)\ge \beta (1)t^{2\theta }.\end{array}$$Then we can obtain that$$\begin{array}{c}\hfill {\mathcal{L}}_{\epsilon }(t{u}_0)\le \frac{t^p}{p}\Vert u_0{\Vert }_{p,\epsilon }^p+\frac{t^q}{q}{\Vert u_0\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q-\frac{\beta (1)}{2}{t}^{2\theta },\end{array}$$which suggests that$$\begin{array}{c}\hfill {\mathcal{L}}_{\epsilon }(t{u}_0)\to -\infty \mathrm{as}\mathrm{t}\to \infty .\end{array}$$Then taking t sufficiently large and $e=t{u}_0,$ we have ${\Vert e\Vert }_{\epsilon }>\zeta$ and ${\mathcal{L}}_{\epsilon }(e)<0.$$\square$

Noting that $\text{supp}(u_0)\subset {\mathrm{\Lambda }}_{\epsilon },$ by Lemma 3.1 we can derive that there exists a constant $B>0$ independent of $\epsilon ,k$ and a such that$$\begin{array}{c}\hfill c_{\epsilon }:=\underset{u\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}\backslash \{0\}}{inf}\underset{t\ge 0}{max}{\mathcal{L}}_{\epsilon }(tu)\le \underset{t\ge 0}{max}{\mathcal{L}}_{\epsilon }(t{u}_0)<B.\end{array}$$Consider the set$$\begin{array}{c}\hfill \mathcal{K}:=\left\{u,\in ,{\mathbb{X}}_{{\mathcal{U}}_{\epsilon }},:,{\Vert u\Vert }_{\epsilon }^p,\le ,\frac{2q\theta }{\theta -q},(B+1)\right\}.\end{array}$$

Lemma 3.2

There exists $k_0>0$ such that$$\begin{array}{c}\hfill \frac{{sup}_{u\in \mathcal{K}}|\frac{1}{{|x|}^{\mu }}\ast G_{\epsilon }(x,u)|}{k_0}<\frac{1}{2}\mathrm{for}\mathrm{each}\epsilon >0.\end{array}$$

Proof

We shall demonstrate that there exists constant $\kappa >0$ such that

3.4

$$\begin{array}{c}\hfill \underset{u\in \mathcal{K}}{sup}|{\int }_{{\mathbb{R}}^N}\frac{G_{\epsilon }(y,u)}{{|x-y|}^{\mu }}\text{d}y|\le \kappa .\end{array}$$In view of $(f_1)$ and $(f_2),$ we conclude that

3.5

$$\begin{array}{c}\hfill |G_{\epsilon }(y,u)|\le C_1{(|u|}^p+{|u|}^{r_0})\mathrm{for}\mathrm{any}\mathrm{y}\in {\mathbb{R}}^{\mathrm{N}}\mathrm{and}\epsilon >0.\end{array}$$Using (3.5), one has that for each $u\in \mathcal{K},$

3.6

$$\begin{array}{cc}\hfill |{\int }_{{\mathbb{R}}^N}\frac{G_{\epsilon }(y,u)}{{|x-y|}^{\mu }}\text{d}y|& \le C_1|{\int }_{{\mathbb{R}}^N}\frac{{|u|}^p+{|u|}^{r_0}}{{|x-y|}^{\mu }}\text{d}y|\hfill \\ \hfill & \le C_1({\int }_{B_1(x)}\frac{{|u|}^p+{|u|}^{r_0}}{{|x-y|}^{\mu }}\text{d}y+{\int }_{B_1^c(x)}\frac{{|u|}^p+{|u|}^{r_0}}{{|x-y|}^{\mu }}\text{d}y)\hfill \\ \hfill & \le C_1({\int }_{B_1(x)}\frac{{|u|}^p+{|u|}^{r_0}}{{|x-y|}^{\mu }}\text{d}y+{\int }_{B_1^c(x)}{(|u|}^p+{|u|}^{r_0})\text{d}y)\hfill \\ \hfill & \le C_2({\int }_{B_1(x)}\frac{{|u|}^p}{{|x-y|}^{\mu }}\text{d}y+{\int }_{B_1(x)}\frac{{|u|}^{r_0}}{{|x-y|}^{\mu }}\text{d}y+1).\hfill \end{array}$$Due to ${\Vert u\Vert }_{\epsilon }\le 2^{q-1}q(\kappa +1),$ applying the Hölder inequality with exponent $t\in (\frac{N}{N-\mu },\frac{N}{N-q}),$ we have

3.7

$$\begin{array}{cc}\hfill {\int }_{B_1(x)}\frac{{|u|}^p}{{|x-y|}^{\mu }}\text{d}y& \le ({\int }_{B_1(x)}{|u|}^{\mathit{tp}}\text{d}y{)}^{\frac{1}{t}}({\int }_{B_1(x)}\frac{1}{{|x-y|}^{\frac{t\mu }{t-1}}}dy{)}^{\frac{t-1}{t}}\hfill \\ \hfill & \le C_3{\Vert u\Vert }_{\epsilon }^p({\int }_{B_1(0)}\frac{1}{{|z|}^{\frac{t\mu }{t-1}}}\text{d}y{)}^{\frac{t-1}{t}}\le C_4,\hfill \end{array}$$where the facts that $t<\frac{N}{N-q}$ implies $tp<q^{\ast }$ and $t>\frac{N}{N-\mu }$ implies $\frac{t\mu }{t-1}<N$ are used. Moreover, by the Hölder inequality with exponent $s\in (\frac{N}{N-\mu },\frac{\mathit{Nq}}{r(N-q)}),$ the fact that $r<\frac{q(N-\mu )}{N-q}$ is equivalent to $\frac{N}{N-\mu }<\frac{\mathit{Nq}}{r(N-q)}$ and ${\Vert u\Vert }_{\epsilon }\le 2^{q-1}q(\kappa +1),$ one can obtain

3.8

$$\begin{array}{cc}\hfill {\int }_{B_1(x)}\frac{{|u|}^{r_0}}{{|x-y|}^{\mu }}\text{d}y& \le ({\int }_{B_1(x)}{|u|}^{s{r}_0}\text{d}y{)}^{\frac{1}{s}}({\int }_{B_1(x)}\frac{1}{{|x-y|}^{\frac{s\mu }{s-1}}}\text{d}y{)}^{\frac{s-1}{s}}\hfill \\ \hfill & \le C_5{\Vert u\Vert }_{\epsilon }^r({\int }_{B_1(0)}\frac{1}{{|z|}^{\frac{s\mu }{s-1}}}\text{d}y{)}^{\frac{s-1}{s}}\le C_6.\hfill \end{array}$$In the light of (3.6)–(3.8), we can infer that (3.4) holds. Let $k_0=2\kappa .$ The proof is finished. $\square$

Define ${\beta }_{\epsilon }(u)={|\mathrm{\nabla }u|}^p+{\mathcal{U}}_{\epsilon }{(x)|\mathrm{\nabla }u|}^q+V_{\epsilon }{(x)(|u|}^p+{\mathcal{U}}_{\epsilon }{(x)|u|}^q)$ for any $u\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}.$ Next, we show a local compactness result for the ${(PS)}_c$ sequence of ${\mathcal{L}}_{\epsilon }.$

Lemma 3.3

For each $c\in [c_{\epsilon },B],$ let $\{u_n\}\subset {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}$ be a ${(PS)}_c$ sequence of ${\mathcal{L}}_{\epsilon }$ at the level $c\in \mathbb{R}.$ Then $\{u_n\}\subset \mathcal{K}$ and for any $\gamma >0,$ there exists $R=R_{\gamma }>0$ being large enough such that$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}{\int }_{B_R^c}{\beta }_{\epsilon }(u_n)\text{d}x\le \gamma .\end{array}$$

Proof

Note that when $\Vert u_n{\Vert }_{\epsilon }\le 2,$ we have $u_n\in \mathcal{K}.$ We shall prove that in the case $\Vert u_n\Vert >2,$ there holds $u_n\in \mathcal{K}.$ Suppose that $\frac{\theta }{p}\le 2.$ According to $(g_3),$ Lemma 3.2, we conclude that

3.9

$$\begin{array}{cc}& \frac{1}{\theta }{\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,u_n)\right)g_{\epsilon }(x,u_n)u_n\text{d}x-\frac{1}{2}{\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,u_n)\right)G_{\epsilon }(x,u_n)\text{d}x\hfill \\ \hfill & =\frac{1}{2\theta }{\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,u_n)\right)(2g_{\epsilon }(x,u_n)u_n-\theta G_{\epsilon }(x,u_n))\text{d}x\hfill \\ \hfill & \ge \frac{1}{2\theta }{\int }_{{\mathrm{\Lambda }}_{\epsilon }^c}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,u_n)\right)\left(2,-,\frac{\theta }{p}\right)g_{\epsilon }(x,u_n)u_n\text{d}x\hfill \\ \hfill & \ge \frac{1}{2\theta }{\int }_{{\mathbb{R}}^N}\frac{k}{2}\left(2,-,\frac{\theta }{p}\right)\frac{V_0}{k}{|u_n|}^p\text{d}x\hfill \\ \hfill & \ge \left(\frac{1}{2\theta },-,\frac{1}{4p}\right){\Vert u_n\Vert }_{p,\epsilon }^p.\hfill \end{array}$$Exploiting the fact that $\{u_n\}$ is a ${(PS)}_c$ sequence of ${\mathcal{L}}_{\epsilon },$ (3.9) and Lemma 2.4, we deduce$$\begin{array}{cc}\hfill B+1& \ge {\mathcal{L}}_{\epsilon }(u_n)-\frac{1}{\theta }⟨{\mathcal{L}}_{\epsilon }^{\prime }(u_n),u_n⟩\hfill \\ \hfill & =\left(\frac{1}{p},-,\frac{1}{\theta }\right)\Vert u_n{\Vert }_{p,\epsilon }^p+\left(\frac{1}{q},-,\frac{1}{\theta }\right){\Vert u_n\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q\hfill \\ \hfill & +\frac{1}{\theta }{\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,u_n)\right)g_{\epsilon }(x,u_n)u_n\text{d}x\hfill \\ \hfill & -\frac{1}{2}{\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,u_n)\right)G_{\epsilon }(x,u_n)\text{d}x\hfill \\ \hfill & \ge \left(\frac{1}{p},-,\frac{1}{\theta }\right)\Vert u_n{\Vert }_{p,\epsilon }^p+\left(\frac{1}{q},-,\frac{1}{\theta }\right)\Vert u_n{\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q+\left(\frac{1}{2\theta },-,\frac{1}{4p}\right){\Vert u_n\Vert }_{p,\epsilon }^p\hfill \\ \hfill & \ge \frac{1}{2}\left(\frac{1}{p},-,\frac{1}{\theta }\right)\Vert u_n{\Vert }_{p,\epsilon }^p+\left(\frac{1}{q},-,\frac{1}{\theta }\right){\Vert u_n\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q\hfill \\ \hfill & \ge \frac{1}{2}\left(\frac{1}{q},-,\frac{1}{\theta }\right)(\Vert u_n{\Vert }_{p,\epsilon }^p+\Vert u_n{\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q)\hfill \\ \hfill & \le \frac{1}{2}\left(\frac{1}{q},-,\frac{1}{\theta }\right){\Vert u_n\Vert }_{\epsilon }^p,\hfill \end{array}$$which shows that $\Vert u_n{\Vert }_{\epsilon }^p\le \frac{2q\theta }{\theta -q}(B+1).$ Thus, we have that if $\Vert u_n{\Vert }_{\epsilon }\ge 2,$ then $u_n\in \mathcal{K}.$ So, we can obtain that for any ${(PS)}_c$ sequence $\{u_n\}$ of ${\mathcal{L}}_{\epsilon },$ one has $\{u_n\}\subset \mathcal{K}.$

For any $R>0,$ we define the function ${\varphi }_R\in C^1({\mathbb{R}}^N)$ fulfilling ${\varphi }_R=0$ in $B_{\frac{R}{2}},$${\varphi }_R=1$ in $B_R^c$ and $|\mathrm{\nabla }{\varphi }_R|\le \frac{4}{R}.$ Applying the fact that $\{{\varphi }_R{u}_n\}$ is bounded in ${\mathbb{X}}_{{\mathcal{U}}_{\epsilon }},$ we can derive that

3.10

$$\begin{array}{c}\hfill ⟨{\mathcal{L}}_{\epsilon }^{\prime }(u_n),{\varphi }_R{u}_n⟩\to 0.\end{array}$$In the light of (3.10), $\{u_n\}\subset \mathcal{K}$ and Lemma 3.2, there holds$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^N}(|\mathrm{\nabla }{u}_n{|}^p+V_{\epsilon }(x)|u_n{|}^p+|\mathrm{\nabla }{u}_n{|}^q+V_{\epsilon }(x)|u_n{|}^q){\varphi }_R\text{d}x\hfill \\ \hfill & =-{\int }_{{\mathbb{R}}^N}{u}_n|\mathrm{\nabla }{u}_n{|}^{p-2}\mathrm{\nabla }{u}_n\mathrm{\nabla }{\varphi }_R\text{d}x-{\int }_{{\mathbb{R}}^N}{u}_n{|\mathrm{\nabla }{u}_n|}^{q-2}\mathrm{\nabla }{u}_n\mathrm{\nabla }{\varphi }_R\text{d}x\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast G_{\epsilon }(x,u_n))g_{\epsilon }(x,u_n)u_n{\varphi }_R\text{d}x+o_n(1)\hfill \\ \hfill & \le {\int }_{{\mathbb{R}}^N}|u_n||\mathrm{\nabla }{u}_n{|}^{p-1}|\mathrm{\nabla }{\varphi }_R|\text{d}x+{\int }_{{\mathbb{R}}^N}|u_n||\mathrm{\nabla }{u}_n{|}^{q-1}|\mathrm{\nabla }{\varphi }_R|\text{d}x\hfill \\ \hfill & +\frac{k}{2}{\int }_{{\mathbb{R}}^N}{g}_{\epsilon }(x,u_n)u_n{\varphi }_R\text{d}x+o_n(1)\hfill \\ \hfill & \le \frac{4}{R}|u_n{|}_p^p|\mathrm{\nabla }{u}_n{|}_p^{p-1}+\frac{4}{R}|u_n{|}_q^q|\mathrm{\nabla }{u}_n{|}_q^{q-1}+\frac{V_0}{2}{\int }_{{\mathbb{R}}^N}{|u_n|}^q{\varphi }_R\text{d}x\hfill \\ \hfill & \le \frac{C_1}{R}+\frac{1}{2}{\int }_{{\mathbb{R}}^N}{V}_{\epsilon }(x){|u_n|}^q{\varphi }_R\text{d}x.\hfill \end{array}$$Thereby, we conclude that$$\begin{array}{cc}& {\int }_{B_R^c}(|\mathrm{\nabla }{u}_n{|}^p+V_{\epsilon }(x)|u_n{|}^p+|\mathrm{\nabla }{u}_n{|}^q+V_{\epsilon }(x)|u_n{|}^q)\text{d}x\hfill \\ \hfill & \le {\int }_{{\mathbb{R}}^N}(|\mathrm{\nabla }{u}_n{|}^p+V_{\epsilon }(x)|u_n{|}^p+|\mathrm{\nabla }{u}_n{|}^q+V_{\epsilon }(x)|u_n{|}^q){\varphi }_R\text{d}x\hfill \\ \hfill & \le \frac{2C_1}{R}\to 0\mathrm{as}\mathrm{R}\to \infty .\hfill \end{array}$$Then for any $\gamma >0,$ taking $R=\frac{2C_1}{\gamma },$ we have$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}{\int }_{B_R^c}{\beta }_{\epsilon }(u_n)\text{d}x\le \gamma .\end{array}$$$\square$

Remark 3.1

From the proof of Lemma 3.3, we can see that any ${(PS)}_c$ sequence of ${\mathcal{L}}_{\epsilon }$ is bounded.

The following point by point gradient convergence of ${(PS)}_c$ sequence is very useful to study the ${(PS)}_c$ condition.

Lemma 3.4

For any $c\in \mathbb{R},$ if $\{u_n\}$ is a ${(PS)}_c$ sequence of ${\mathcal{L}}_{\epsilon }.$ Then we have that $\mathrm{\nabla }{u}_n\to \mathrm{\nabla }u$ a.e. in ${\mathbb{R}}^N.$

Proof

By virtue of Remark 3.1 and Lemma 2.3, we can infer that there exists $u\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}$ such that

3.11

$$\begin{array}{c}\hfill u_n\rightharpoonup u\mathrm{in}{\mathbb{X}}_{{\mathcal{U}}_{\epsilon }},{\mathrm{u}}_{\mathrm{n}}\to \mathrm{u}\mathrm{in}{\mathrm{L}}_{\mathrm{loc}}^{\mathrm{r}}({\mathbb{R}}^{\mathrm{N}})\mathrm{and}{\mathrm{u}}_{\mathrm{n}}\to \mathrm{u}\mathrm{a}.\mathrm{e}.\mathrm{in}{\mathbb{R}}^{\mathrm{N}}.\end{array}$$Let $R>0$ satisfy ${\mathrm{\Lambda }}_{\epsilon }\subset B_R.$ Define ${\psi }_R\in C_c^1({\mathbb{R}}^N)$ such that ${\psi }_R=1$ in $B_R,$${\psi }_R=0$ in $B_{2R}^c$ and $|\mathrm{\nabla }{\psi }_R|\le \frac{2}{R}.$ Notice that $\{(u_n-u){\psi }_R\}$ is bounded in ${\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}.$ Then we deduce that $⟨{\mathcal{L}}_{\epsilon }^{\prime }(u_n),(u_n-u){\psi }_R⟩\to 0.$ This suggests that

3.12

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^{p-2}\mathrm{\nabla }{u}_n(\mathrm{\nabla }{u}_n-\mathrm{\nabla }u){\psi }_R\text{d}x+{\int }_{{\mathbb{R}}^N}(u_n-u){|\mathrm{\nabla }{u}_n|}^{p-2}\mathrm{\nabla }{u}_n\mathrm{\nabla }{\psi }_R\text{d}x\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^N}{\mathcal{U}}_{\epsilon }(x){|\mathrm{\nabla }{u}_n|}^{q-2}\mathrm{\nabla }{u}_n(\mathrm{\nabla }{u}_n-\mathrm{\nabla }u){\psi }_R\text{d}x\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^N}{\mathcal{U}}_{\epsilon }(x)(u_n-u){|\mathrm{\nabla }{u}_n|}^{q-2}\mathrm{\nabla }{u}_n\mathrm{\nabla }{\psi }_R\text{d}x\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^N}{V}_{\epsilon }(x)(|u_n{|}^{p-2}{u}_n+{\mathcal{U}}_{\epsilon }(x)|u_n{|}^{q-2}{u}_n)(u_n-u){\psi }_R\text{d}x\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast G_{\epsilon }(x,u_n))g_{\epsilon }(x,u_n)(u_n-u){\psi }_R\text{d}x.\hfill \end{array}$$Using Lemma 3.2, $(f_1),$$(f_2),$ Hölder inequality and (3.11), we conclude that

3.13

$$\begin{array}{cc}& |{\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,u_n)\right)g_{\epsilon }(x,u_n)(u_n-u){\psi }_R\text{d}x|\hfill \\ \hfill & \le \frac{k}{2}{\int }_{{\mathbb{R}}^N}|g_{\epsilon }(x,u_n)(u_n-u){\psi }_R|\text{d}x\hfill \\ \hfill & \le \frac{k}{2}\frac{2}{R}{\int }_{B_{2R}}{C}_1(|u_n{|}^{p-1}+|u_n{|}^{r_0-1})|u_n-u|\text{d}x\hfill \\ \hfill & \le C_2({\int }_{B_{2R}}|u_n{|}^p\text{d}x{)}^{\frac{p-1}{p}}({\int }_{B_{2R}}{|u_n-u|}^p\text{d}x{)}^{\frac{1}{p}}\hfill \\ \hfill & +C_2({\int }_{B_{2R}}|u_n{|}^{r_0}\text{d}x{)}^{\frac{r_0-1}{r_0}}({\int }_{B_{2R}}{|u_n-u|}^{r_0}\text{d}x{)}^{\frac{1}{r_0}}\hfill \\ \hfill & \to 0.\hfill \end{array}$$Similarly, we can obtain that

3.14

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}(u_n-u){|\mathrm{\nabla }{u}_n|}^{p-2}\mathrm{\nabla }{u}_n\mathrm{\nabla }{\psi }_R\text{d}x\to 0,\end{array}$$

3.15

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}{\mathcal{U}}_{\epsilon }(x)(u_n-u){|\mathrm{\nabla }{u}_n|}^{q-2}\mathrm{\nabla }{u}_n\mathrm{\nabla }{\psi }_R\text{d}x\to 0\end{array}$$and

3.16

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}{V}_{\epsilon }(x)(|u_n{|}^{p-2}{u}_n+{\mathcal{U}}_{\epsilon }(x)|u_n{|}^{q-2}{u}_n)(u_n-u){\psi }_R\text{d}x\to 0.\end{array}$$Combining with (3.12)–(3.16), one can infer that

3.17

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^{p-2}\mathrm{\nabla }{u}_n(\mathrm{\nabla }{u}_n-\mathrm{\nabla }u){\psi }_R\text{d}x+{\int }_{{\mathbb{R}}^N}{\mathcal{U}}_{\epsilon }(x){|\mathrm{\nabla }{u}_n|}^{q-2}\mathrm{\nabla }{u}_n(\mathrm{\nabla }{u}_n-\mathrm{\nabla }u){\psi }_R\text{d}x\to 0.\end{array}$$Moreover, [11, Lemma 2.1] shows that for any $\zeta ,\eta \in {\mathbb{R}}^N,$

3.18

$$\begin{array}{c}\hfill {(|\zeta |}^{r-2}\zeta -{|\eta |}^{r-2}\eta )(\zeta -\eta )\ge d_1{|\zeta -\eta |}^r\mathrm{for}\mathrm{each}\mathrm{r}\ge 2,\end{array}$$and

3.19

$$\begin{array}{c}\hfill {(|\zeta |}^{r-2}\zeta -{|\eta |}^{r-2}\eta )(\zeta -\eta )\ge d_2(|\zeta |+{|\eta |)}^{r-2}{|\zeta -\eta |}^2\mathrm{for}\mathrm{each}\mathrm{r}\in (1,2).\end{array}$$Besides, we can infer that if $r\in (1,2),$ there holds that for any $\zeta ,\eta \in {\mathbb{R}}^N,$$$\begin{array}{c}\hfill (|\zeta |+{|\eta |)}^r\le 2^{r-1}{(|\zeta |}^r+{|\eta |}^r).\end{array}$$Thus for each $r\in (1,2),$ we can use (3.19) to see that

3.20

$$\begin{array}{c}\hfill {[(|\zeta |}^{r-2}\zeta -{|\eta |}^{r-2}{\eta )(\zeta -\eta )]}^{\frac{r}{2}}{(|\zeta |}^r+{|\eta |}^r{)}^{\frac{2-r}{2}}\ge d_3{|\zeta -\eta |}^r\mathrm{for}\mathrm{any}\zeta ,\eta \in {\mathbb{R}}^{\mathrm{N}}.\end{array}$$By (3.20), the Hölder inequality, (3.17) and the boundedness of $\{u_n\}$ in ${\mathbb{X}}_{{\mathcal{U}}_{\epsilon }},$ we have that

3.21

$$\begin{array}{cc}& {\int }_{B_R}{|\mathrm{\nabla }{u}_n-\mathrm{\nabla }u|}^p\text{d}x\hfill \\ \hfill & \le C{\int }_{B_R}[(|\mathrm{\nabla }{u}_n{|}^{p-2}\mathrm{\nabla }{u}_n-{|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u)(\mathrm{\nabla }{u}_n-\mathrm{\nabla }u){]}^{\frac{p}{2}}(|\mathrm{\nabla }{u}_n{|}^p\hfill \\ \hfill & +{|\mathrm{\nabla }u|}^p{)}^{\frac{2-p}{2}}\text{d}x\hfill \\ \hfill & \le C({\int }_{B_R}(|\mathrm{\nabla }{u}_n{|}^{p-2}\mathrm{\nabla }{u}_n-{|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u)(\mathrm{\nabla }{u}_n-\mathrm{\nabla }u)\text{d}x{)}^{\frac{p}{2}}\hfill \\ \hfill & ({\int }_{B_R}(|\mathrm{\nabla }{u}_n{|}^p+{|\mathrm{\nabla }u|}^p)\text{d}x{)}^{\frac{2-p}{2}}\hfill \\ \hfill & =o_n(1).\hfill \end{array}$$When $p\ge 2,$ we can exploit (3.17) and (3.18) to obtain that

3.22

$$\begin{array}{c}\hfill {\int }_{B_{R^{\prime }}}|\mathrm{\nabla }{u}_n{-\mathrm{\nabla }u|}^p\text{d}x\le {\int }_{B_{R^{\prime }}}(|\mathrm{\nabla }{u}_n{|}^{p-2}\mathrm{\nabla }{u}_n-{|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u)(\mathrm{\nabla }{u}_n-\mathrm{\nabla }u)\text{d}x=o_n(1).\end{array}$$It follows from (3.14) and (3.15) that ${\int }_{B_R}{|u_n-u|}^p\text{d}x\to 0.$ Since R is arbitrary, one can derive that$$\begin{array}{c}\hfill \mathrm{\nabla }{u}_n\to \mathrm{\nabla }u\mathrm{a}.\mathrm{e}.\mathrm{in}{\mathbb{R}}^{\mathrm{N}}.\end{array}$$$\square$

Lemma 3.5

${\mathcal{L}}_{\epsilon }$ fulfills the ${(PS)}_c$ condition for each $c\in [c_{\epsilon },B].$

Proof

Let $\{u_n\}$ be a ${(PS)}_c$ sequence. Then up to a subsequence there exists $u\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}$ such that

3.23

$$\begin{array}{c}\hfill u_n\rightharpoonup u\mathrm{in}{\mathbb{X}}_{{\mathcal{U}}_{\epsilon }},{\mathrm{u}}_{\mathrm{n}}\to \mathrm{u}\mathrm{in}{\mathrm{L}}_{\mathrm{loc}}^{\mathrm{r}}({\mathbb{R}}^{\mathrm{N}})\mathrm{and}{\mathrm{u}}_{\mathrm{n}}\to \mathrm{u}\mathrm{a}.\mathrm{e}.\mathrm{in}{\mathbb{R}}^{\mathrm{N}}.\end{array}$$Notice that $⟨{\mathcal{L}}_{\epsilon }^{\prime }(u_n),u_n⟩=o_n(1).$ Then we have

3.24

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}{\beta }_{\epsilon }(u_n)\text{d}x={\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast G_{\epsilon }(x,u_n))g_{\epsilon }(x,u_n)u_n\text{d}x.\end{array}$$By Lemmas 3.4 and 3.3 and (3.23), there holds that

3.25

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^N}({|\mathrm{\nabla }{u}_n|}^{p-2}\mathrm{\nabla }{u}_n\mathrm{\nabla }u\hfill \\ \hfill & +{\mathcal{U}}_{\epsilon }(x)|\mathrm{\nabla }{u}_n{|}^{q-2}\mathrm{\nabla }{u}_n\mathrm{\nabla }u+V_{\epsilon }(x)(|u_n{|}^{p-2}{u}_{n}u+{\mathcal{U}}_{\epsilon }(x)|u_n{|}^{q-2}{u}_{n}u))\text{d}x\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}{\beta }_{\epsilon }(u)\text{d}x+o_n(1).\hfill \end{array}$$It follows from $⟨{\mathcal{L}}_{\epsilon }^{\prime }(u_n),u⟩=o_n(1)$ and (3.25) that

3.26

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}{\beta }_{\epsilon }(u)\text{d}x={\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast G_{\epsilon }(x,u_n))g_{\epsilon }(x,u_n)u\text{d}x.\end{array}$$From Lemma 3.3, we know that $\{u_n\}\subset \mathcal{K}.$ By virtue of $\{u_n\}\subset \mathcal{K},$ Lemma 3.2, ${\mathrm{\Lambda }}_{\epsilon }\subset B_R$ and $(g_3),$ one concludes that

3.27

$$\begin{array}{cc}& |{\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast G_{\epsilon }(x,u_n))g_{\epsilon }(x,u_n)(u_n-u)\text{d}x|\hfill \\ \hfill & \le \frac{k}{2}{\int }_{{\mathbb{R}}^N}|g_{\epsilon }(x,u_n)(u_n-u)|\text{d}x\hfill \\ \hfill & =\frac{k}{2}{\int }_{B_R}|g_{\epsilon }(x,u_n)(u_n-u)|\text{d}x+\frac{k}{2}{\int }_{B_R^c}|g_{\epsilon }(x,u_n)(u_n-u)|\text{d}x\hfill \\ \hfill & \le C{\int }_{B_R}(|u_n{|}^{p-1}+|u_n{|}^{r_0-1})|u_n-u|\text{d}x+\frac{k}{2}{\int }_{B_R^c}\frac{V_0}{k}|u_n{|}^{p-1}|u_n-u|\text{d}x.\hfill \end{array}$$Furthermore, one can derive from the Hölder inequality and (3.23) that

3.28

$$\begin{array}{c}\hfill {\int }_{B_R}(|u_n{|}^{p-1}+|u_n{|}^{r_0-1})|u_n-u|\text{d}x\to 0.\end{array}$$In view of Lemma 3.3, the Hölder inequality and the boundedness of $\{u_n\}$ in ${\mathbb{X}}_{{\mathcal{U}}_{\epsilon }},$ we can derive

3.29

$$\begin{array}{cc}& V_0{\int }_{B_R^c}|u_n{|}^{p-1}|u_n-u|\text{d}x\le V_0{\int }_{B_R^c}|u_n{|}^p\text{d}x+V_0{\int }_{B_R^c}|u_n{|}^{p-1}|u|\text{d}x\hfill \\ \hfill & \le {\int }_{B_R^c}{\beta }_{\epsilon }(u_n)\text{d}x+V_0({\int }_{B_R^c}|u_n{|}^p\text{d}x{)}^{\frac{p-1}{p}}({\int }_{B_R^c}{|u|}^p\text{d}x{)}^{\frac{1}{p}}\le \gamma +o_n(1).\hfill \end{array}$$According to (3.27)–(3.29), one has that

3.30

$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,u_n)\right)g_{\epsilon }(x,u_n)(u_n-u)\text{d}x\le \frac{\gamma }{2}+o_n(1).\end{array}$$Due to the arbitrariness of $\gamma ,$ (3.24) and (3.26), we can infer$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}{\beta }_{\epsilon }(u_n)\text{d}x\to {\int }_{{\mathbb{R}}^N}{\beta }_{\epsilon }(u)\text{d}x,\end{array}$$which suggests that $u_n\to u$ in ${\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}.$$\square$

Let ${\mathcal{S}}_{\epsilon }$ denote the unite sphere in ${\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}.$ Then ${\mathcal{S}}_{\epsilon }$ is a $C^{1,1}$ manifold of codimension one. This is$$\begin{array}{c}\hfill {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}=T_u{\mathcal{S}}_{\epsilon }\oplus \mathbb{R}u,\end{array}$$where$$\begin{array}{cc}\hfill T_u{\mathcal{S}}_{\epsilon }& =\{v\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}:{\int }_{{\mathbb{R}}^N}{((|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u+{\mu }_{\epsilon }{(x)|\mathrm{\nabla }u|}^{q-2}\mathrm{\nabla }u)\mathrm{\nabla }v\hfill \\ \hfill & +V_{\epsilon }{(x)(|u|}^{p-2}u+{\mu }_{\epsilon }{(x)|u|}^{q-2}u)v)\text{d}x=0\}.\hfill \end{array}$$Next we set up a contact between ${\mathcal{S}}_{\epsilon }$ and ${\mathcal{N}}_{\epsilon }.$

Lemma 3.6

There holds that

1. for any $u\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }},$ let $h_u:[0,\infty )\to \mathbb{R}$ be defined as $h_u(t):={\mathcal{L}}_{\epsilon }(tu).$ We have that there is a unique number $t_u>0$ fulfilling $h_u^{\prime }(t)>0$ in $(0,t_u),$ and $h_u^{\prime }(t)<0$ in $(t_u,+\infty );$

2. there is $\tau >0,$ such that $t_u\ge \tau$ for every $u\in {\mathcal{S}}_{\epsilon }.$ Furthermore,  if $\mathcal{K}\subset {\mathcal{S}}_{\epsilon }$ is a compact set,  then there exists $C_{\mathcal{K}}>0$ only dependent on $\mathcal{K}$ such that $t_u\le C_{\mathcal{K}}$ for each $u\in \mathcal{K};$

3. set ${\widehat{m}}_{\epsilon }:{\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}\to {\mathcal{N}}_{\epsilon }$ be defined as ${\widehat{m}}_{\epsilon }(u):=t_{u}u.$ Then ${\widehat{m}}_{\epsilon }$ is continuous and $m_{\epsilon }:={\widehat{m}}_{\epsilon }{|}_{{\mathcal{S}}_{\epsilon }}$ is a homeomorphism between ${\mathcal{S}}_{\epsilon }$ and ${\mathcal{N}}_{\epsilon }.$ Moreover,  $m_{\epsilon }^{-1}(u)=\frac{u}{{\Vert u\Vert }_{\epsilon }};$

Proof

(i) It follows from Lemma 3.1 that for any $u\in {\mathcal{S}}_{\epsilon },$ there holds

3.31

$$\begin{array}{c}\hfill h_u(t)\to 0^{+}\mathrm{as}\mathrm{t}\to 0^{+}\mathrm{and}{\mathrm{h}}_{\mathrm{u}}(\mathrm{t})\to -\infty \mathrm{as}\mathrm{t}\to \infty .\end{array}$$Using (3.31), we conclude that there exists $t_u\in (0,\infty )$ such that

3.32

$$\begin{array}{c}\hfill h_u(t_u)=\underset{t\in (0,\infty )}{max}{h}_u(t)\mathrm{and}{\mathrm{h}}_{\mathrm{u}}^{\prime }({\mathrm{t}}_{\mathrm{u}})=0.\end{array}$$It suffices to show that $t_u$ is the unique number such that (3.32) holds. In contrary, we suppose that there exists $t_0\in (0,\infty )$ such that

3.33

$$\begin{array}{c}\hfill h_u(t_0)=\underset{t\in (0,\infty )}{max}{h}_u(t)\mathrm{and}{\mathrm{h}}_{\mathrm{u}}^{\prime }({\mathrm{t}}_0)=0.\end{array}$$Without the loss of generality, assume $t_0<t_u.$ Then by (3.32) and (3.33), we have$$\begin{array}{c}\hfill t_u^{p-1}{\Vert u\Vert }_{p,\epsilon }^p+t_u^{q-1}{\Vert u\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q={\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,t_{u}u)\right)g_{\epsilon }(x,t_{u}u)u\text{d}x\end{array}$$and$$\begin{array}{c}\hfill t_0^{p-1}{\Vert u\Vert }_{p,\epsilon }^p+t_0^{q-1}{\Vert u\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q={\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,t_{0}u)\right)g_{\epsilon }(x,t_{0}u)u\text{d}x.\end{array}$$Thus exploiting $(g_4),$ we deduce that$$\begin{array}{cc}& \left(\frac{1}{t_0^{q-p}},-,\frac{1}{t_u^{q-p}}\right){\Vert u\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,t_{0}u)\right)\frac{g_{\epsilon }(x,t_{0}u)u}{t_0^{q-1}}\text{d}x\hfill \\ \hfill & -{\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,t_{u}u)\right)\frac{g_{\epsilon }(x,t_{u}u)u}{t_u^{q-1}}\text{d}x\hfill \\ \hfill & \le {\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,t_{u}u)\right)(\frac{g_{\epsilon }(x,t_{0}u)}{{(t_{0}u)}^{q-1}}-\frac{g_{\epsilon }(x,t_{u}u)}{{(t_{u}u)}^{q-1}})u^q\text{d}x\hfill \\ \hfill & \le {\int }_{{\mathrm{\Lambda }}_{\epsilon }^c}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,t_{u}u)\right)(\frac{g_{\epsilon }(x,t_{0}u)}{{(t_{0}u)}^{q-1}}-\frac{g_{\epsilon }(x,t_{u}u)}{{(t_{u}u)}^{q-1}})u^q\text{d}x\hfill \\ \hfill & ={\int }_{{\mathrm{\Lambda }}_{\epsilon }^c\cap \{t_{u}u\le a\}}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,t_{u}u)\right)(\frac{g_{\epsilon }(x,t_{0}u)}{{(t_{0}u)}^{q-1}}-\frac{g_{\epsilon }(x,t_{u}u)}{{(t_{u}u)}^{q-1}})u^q\text{d}x\hfill \\ \hfill & +{\int }_{{\mathrm{\Lambda }}_{\epsilon }^c\cap \{t_{u}u>a\}\cap \{t_{0}u\ge a\}}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,t_{u}u)\right)(\frac{g_{\epsilon }(x,t_{0}u)}{{(t_{0}u)}^{q-1}}-\frac{g_{\epsilon }(x,t_{u}u)}{{(t_{u}u)}^{q-1}})u^q\text{d}x\hfill \\ \hfill & +{\int }_{{\mathrm{\Lambda }}_{\epsilon }^c\cap \{t_{u}u>a\}\cap \{t_{0}u<a\}}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,t_{u}u)\right)(\frac{g_{\epsilon }(x,t_{0}u)}{{(t_{0}u)}^{q-1}}-\frac{g_{\epsilon }(x,t_{u}u)}{{(t_{u}u)}^{q-1}})u^q\text{d}x\hfill \\ \hfill & \le {\int }_{{\mathrm{\Lambda }}_{\epsilon }^c\cap \{t_{u}u>a\}\cap \{t_{0}u<a\}}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,t_{u}u)\right)(\frac{g_{\epsilon }(x,t_{0}u)}{{(t_{0}u)}^{q-1}}-\frac{g_{\epsilon }(x,t_{u}u)}{{(t_{u}u)}^{q-1}})u^q\text{d}x\hfill \\ \hfill & \le {\int }_{{\mathrm{\Lambda }}_{\epsilon }^c\cap \{t_{u}u>a\}\cap \{t_{0}u<a\}}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,t_{u}u)\right)(\frac{f(a)}{a^{q-1}}-\frac{V_0}{k})u^q\text{d}x\hfill \\ \hfill & ={\int }_{{\mathrm{\Lambda }}_{\epsilon }^c\cap \{t_{u}u>a\}\cap \{t_{0}u<a\}}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{\epsilon },(x,t_{u}u)\right)(\frac{V_0}{k}-\frac{V_0}{k})u^q\text{d}x=0.\hfill \end{array}$$It is impossible due to $u\ne 0$ in ${\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}.$

(ii) $h_u^{\prime }(t_u)=0$ implies that

3.34

$$\begin{array}{c}\hfill t_u^p{\Vert u\Vert }_{p,\epsilon }^p+t_u^q{\Vert u\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q={\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast G_{\epsilon }(x,t_{u}u))g_{\epsilon }(x,t_{u}u)t_{u}u\text{d}x.\end{array}$$By Lemma 2.5, $(f_1),$$(f_2),$$r_0<p_{\mu }^{\ast }$ and Lemma 2.3, we conclude that for any $v\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }},$

3.35

$$\begin{array}{cc}& |{\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast G_{\epsilon }(x,v))g_{\epsilon }(x,v)v\text{d}x|\le C_1|G_{\epsilon }{(x,v)|}_{\frac{2N}{2N-\mu }}{|g_{\epsilon }(x,v)v|}_{\frac{2N}{2N-\mu }}\hfill \\ \hfill & \le C_2{||v|}^p+{|v|}^{r_0}{|}_{\frac{2N}{2N-\mu }}^2\le C_3{(|v|}_{\frac{2Np}{2N-\mu }}^p+{|v|}_{\frac{2N{r}_0}{2N-\mu }}^{r_0}{)}^2\hfill \\ \hfill & \le C_4{(|v|}_{\frac{2Np}{2N-\mu }}^{2p}+{|v|}_{\frac{2N{r}_0}{2N-\mu }}^{2r_0})\le C_5{(\Vert v\Vert }_{\epsilon }^{2p}+{\Vert v\Vert }_{\epsilon }^{2r_0}).\hfill \end{array}$$According to (3.34), (3.35) and Lemma 2.4, one derives that$$\begin{array}{c}\hfill t_u\ge 1\mathrm{or}\frac{1}{\mathrm{q}}{\mathrm{t}}_{\mathrm{u}}^{\mathrm{q}}{\Vert \mathrm{u}\Vert }_{\epsilon }^{\mathrm{q}}\le {\mathrm{t}}_{\mathrm{u}}^{\mathrm{p}}{\Vert \mathrm{u}\Vert }_{\mathrm{p},\epsilon }^{\mathrm{p}}+{\mathrm{t}}_{\mathrm{u}}^{\mathrm{q}}{\Vert \mathrm{u}\Vert }_{\mathrm{q},{\mathcal{U}}_{\epsilon }}^{\mathrm{q}}\le {\mathrm{C}}_5{(\Vert \mathrm{v}\Vert }_{\epsilon }^{2\mathrm{p}}+{\Vert \mathrm{v}\Vert }_{\epsilon }^{2{\mathrm{r}}_0}),\end{array}$$which lead to that there exists $\tau >0$ such that $t_u\ge \tau$ for any $u\in {\mathcal{S}}_{\epsilon }$ due to $q<2p.$

In contrary, assume that there exists a sequence $\{u_n\}\subset \mathcal{K}$ such that $u_n\to u$ in ${\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}$ with $u\in \mathcal{K}.$ So, there holds that$$\begin{array}{c}\hfill t_n:=t_{u_n}\to \infty .\end{array}$$Proceeding as Lemma 3.1, we can conclude that

3.36

$$\begin{array}{c}\hfill {\mathcal{L}}_{\epsilon }(t_n{u}_n)\to -\infty .\end{array}$$Noting that $⟨{\mathcal{L}}_{\epsilon }^{\prime }(t_n{u}_n),t_n{u}_n⟩=0,$ we can use $(g_3)$ to see that

3.37

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\epsilon }(t_n{u}_n)& ={\mathcal{L}}_{\epsilon }(t_n{u}_n)-\frac{1}{2p}⟨{\mathcal{L}}_{\epsilon }^{\prime }(t_n{u}_n),t_n{u}_n⟩\hfill \\ \hfill & =\frac{1}{2p}{t}_n^p\Vert u_n{\Vert }_{p,\epsilon }^p+(\frac{1}{q}-\frac{1}{2p})t_n^q{\Vert u_n\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q\hfill \\ \hfill & +\frac{1}{2p}{\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast G_{\epsilon }(x,t_n{u}_n))(g_{\epsilon }(x,u_n)-p{G}_{\epsilon }(x,t_n{u}_n))\text{d}x\hfill \\ \hfill & \ge \frac{1}{2p}{t}_n^p\Vert u_n{\Vert }_{p,\epsilon }^p+(\frac{1}{q}-\frac{1}{2p})t_n^q{\Vert u_n\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q.\hfill \end{array}$$By (3.37) and the facts that $t_n\to \infty$ and $u_n\to u$ in ${\mathbb{X}}_{{\mathcal{U}}_{\epsilon }},$ one deduces$$\begin{array}{c}\hfill {\mathcal{L}}_{\epsilon }(t_n{u}_n)\to \infty ,\end{array}$$which is false due to (3.36).

(iii) Similar to [33, Lemma 2.7-(iii)], the assertion can be obtained. $\square$

According to Lemma 3.6 and [23, Corollary 10], the following abstract critical point theorem holds.

Proposition 3.1

If $(A_1)$–$(A_4)$ and $(f_1)$–$(f_4)$ hold,  then

1. ${\widehat{\phi }}_{\epsilon }\in C^1({\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}^{+},\mathbb{R})$ and $$\begin{array}{c}\hfill ⟨{\widehat{\phi }}_{\epsilon }^{\prime }(u),v⟩=\frac{\Vert {\widehat{m}}_{\epsilon }{(u)\Vert }_{\epsilon }}{{\Vert u\Vert }_{\epsilon }}⟨{\mathcal{L}}_{\epsilon }^{\prime }({\widehat{m}}_{\epsilon }(u)),v⟩\mathrm{for}\mathrm{every}\mathrm{u}\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}^{+}\mathrm{and}\mathrm{v}\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }};\end{array}$$

2. ${\phi }_{\epsilon }\in C^1({\mathcal{S}}_{\epsilon },\mathbb{R})$ and $⟨{\phi }_{\epsilon }^{\prime }(u),v⟩={\Vert m_{\epsilon }(u)\Vert }_{\epsilon }⟨{\mathcal{L}}_{\epsilon }^{\prime }(m_{\epsilon }(u)),v⟩$ for every $v\in T_u{\mathcal{S}}_{\epsilon };$

3. assume that $\{u_n\}$ is a ${(PS)}_d$ sequence to ${\psi }_{\epsilon },$ we have that $\{m_{\epsilon }(u_n)\}$ is a ${(PS)}_d$ sequence to ${\mathcal{L}}_{\epsilon }.$ Moreover,  assume that $\{u_n\}\subset {\mathcal{N}}_{\epsilon }$ is a bounded ${(PS)}_d$ sequence for ${\mathcal{L}}_{\epsilon }.$ We have that $\{m_{\epsilon }^{-1}(u_n)\}$ is a ${(PS)}_d$ sequence for ${\phi }_{\epsilon };$

4. u is a critical point of ${\phi }_{\epsilon }$ if and only if $m_{\epsilon }(u)$ is a nontrivial critical point of ${\mathcal{L}}_{\epsilon }.$ Moreover,  the corresponding critical value coincides and $$\begin{array}{c}\hfill \underset{u\in {\mathcal{S}}_{\epsilon }}{inf}{\phi }_{\epsilon }(u)=\underset{u\in {\mathcal{N}}_{\epsilon }}{inf}{\mathcal{L}}_{\epsilon }(u).\end{array}$$

Proposition 3.2

${\phi }_{\epsilon }$ fulfills the ${(PS)}_c$ condition for each $c\in [c_{\epsilon },B].$

Proof

Let $\{v_n\}$ be a ${(PS)}_c$ sequence of ${\phi }_{\epsilon }.$ Take $u_n=m_{\epsilon }(v_n).$ By Proposition 4.1, we know that $\{u_n\}$ is a ${(PS)}_c$ sequence of ${\mathcal{L}}_{\epsilon }.$ Due to $c\in [c_{\epsilon },B]$ and Lemma 3.5, there exists $u\in {\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}$ such that $u_n\to u$ in ${\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}.$ Observe that $\{u_n\}\subset {\mathcal{N}}_{\epsilon }.$ Then $u\in {\mathcal{N}}_{\epsilon }.$ Let $v=m_{\epsilon }^{-1}(u).$ Since $u_n\to u$ in ${\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}$ and Lemma 3.6-(iii), one has that$$\begin{array}{c}\hfill v_n=m_{\epsilon }^{-1}(u_n)\to m_{\epsilon }^{-1}(u)=v\mathrm{in}{\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}.\end{array}$$The proof is completed. $\square$

Limit problem

The limit equation of Eq. (1.1) is

4.1

$$\begin{array}{cc}& -{\mathrm{div}(|\mathrm{\nabla }\mathrm{u}|}^{\mathrm{p}-2}\mathrm{\nabla }\mathrm{u}+{\mathcal{U}}_0{|\mathrm{\nabla }\mathrm{u}|}^{\mathrm{q}-2}\mathrm{\nabla }\mathrm{u})+{\mathrm{V}}_0{(|\mathrm{u}|}^{\mathrm{p}-2}\mathrm{u}+{\mathcal{U}}_0{|\mathrm{u}|}^{\mathrm{q}-2}\mathrm{u})\hfill \\ \hfill & =\left(\frac{1}{{|x|}^{\mu }},,\ast ,F,(u)\right)f(u)\mathrm{in}{\mathbb{R}}^{\mathrm{N}},\hfill \end{array}$$where $V_0>0$ and ${\mathcal{U}}_0\ge 0$ Equation (4.1) is settled in the Sobolev space$$\begin{array}{c}\hfill {\mathbb{E}}_{{\mathcal{U}}_0}=\{u:{\mathbb{R}}^N\to \mathbb{R}\mathrm{is}\mathrm{measurable}:\mathrm{u}\in {\mathrm{W}}^{1,\mathrm{p}}({\mathbb{R}}^{\mathrm{N}}),{\mathcal{U}}_0{\int }_{{\mathbb{R}}^{\mathrm{N}}}{(|\mathrm{\nabla }\mathrm{u}|}^{\mathrm{q}}+{|\mathrm{u}|}^{\mathrm{q}})\text{d}\mathrm{x}<\infty \},\end{array}$$which is equipped with the norm$$\begin{array}{c}\hfill {\Vert u\Vert }_{{\mathcal{U}}_0}={\Vert u\Vert }_p+{\Vert u\Vert }_{q,{\mathcal{U}}_0},\end{array}$$where$$\begin{array}{cc}& {\Vert u\Vert }_p=({\int }_{{\mathbb{R}}^N}{(|\mathrm{\nabla }u|}^p+V_0{|u|}^p)\text{d}x{)}^{\frac{1}{p}}\mathrm{and}\hfill \\ \hfill & {\Vert u\Vert }_{q,{\mathcal{U}}_0}={\mathcal{U}}_0({\int }_{{\mathbb{R}}^N}{(|\mathrm{\nabla }u|}^q+V_0{|u|}^q)\text{d}x{)}^{\frac{1}{q}}.\hfill \end{array}$$The variational functional of Eq. (4.1) is given as$$\begin{array}{c}\hfill {\mathcal{J}}_{{\mathcal{U}}_0}(u)=\frac{1}{p}{\Vert u\Vert }_p^p+\frac{1}{q}{\Vert u\Vert }_{q,{\mathcal{U}}_0}^q-{\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast F(u))F(u)\text{d}x.\end{array}$$Then ${\mathcal{J}}_{{\mathcal{U}}_0}$ is differentiable and its differential for any $u,v\in {\mathbb{E}}_{{\mathcal{U}}_0}$ is$$\begin{array}{cc}\hfill ⟨{\mathcal{J}}_{{\mathcal{U}}_0}^{\prime }(u),v⟩& ={\int }_{{\mathbb{R}}^N}{(|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u\mathrm{\nabla }v+V_0{|u|}^{p-2}uv)\text{d}x\hfill \\ \hfill & +{\mu }_0{\int }_{{\mathbb{R}}^N}{(|\mathrm{\nabla }u|}^{q-2}\mathrm{\nabla }u\mathrm{\nabla }v+V_0{|u|}^{q-2}uv)\text{d}x\hfill \\ \hfill & -{\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast F(u))f(u)v\text{d}x.\hfill \end{array}$$Similar to the above section, we have the following several results.

Lemma 4.1

${\mathcal{J}}_{{\mathcal{U}}_0}$ fulfills that

1. there exist $c_0,\zeta >0$ such that ${\mathcal{J}}_{{\mathcal{U}}_0}(u)\ge c_0$ for any $u\in {\mathbb{E}}_{{\mathcal{U}}_0}$ with ${\Vert u\Vert }_{{\mu }_0}=\zeta ;$

2. there exists $e\in {\mathbb{E}}_{{\mathcal{U}}_0}$ with ${\Vert e\Vert }_{{\mu }_0}>\kappa$ such that ${\mathcal{J}}_{{\mathcal{U}}_0}(e)<0.$

Lemma 4.2

Suppose that $\{u_n\}\subset {\mathbb{E}}_{{\mathcal{U}}_0}$ is a Palais–Smale sequence of ${\mathcal{J}}_{{\mathcal{U}}_0}$ at the level $c\in \mathbb{R}.$ Then $\{u_n\}$ is bounded in ${\mathbb{E}}_{{\mathcal{U}}_0}$ and up to a subsequence $\mathrm{\nabla }{u}_n\to \mathrm{\nabla }u$ a.e. in ${\mathbb{R}}^N.$

The Nehari manifold of Eq. (4.1) is defined by$$\begin{array}{c}\hfill {\mathcal{M}}_{{\mathcal{U}}_0}=\{u\in {\mathbb{E}}_{{\mathcal{U}}_0}\backslash \{0\}:⟨{\mathcal{J}}_{{\mathcal{U}}_0}^{\prime }(u),u⟩=0\}.\end{array}$$Let $d_{{\mu }_0}={inf}_{u\in {\mathcal{M}}_{{\mathcal{U}}_0}}{\mathcal{J}}_{{\mathcal{U}}_0}(u)$ and ${\mathcal{S}}_{{\mathcal{U}}_0}$ be the unite sphere in ${\mathbb{E}}_{{\mathcal{U}}_0}.$ We have that ${\mathcal{S}}_{{\mathcal{U}}_0}$ is a $C^{1,1}$ manifold of codimension one. Then ${\mathbb{E}}_{{\mathcal{U}}_0}=T_u{\mathcal{S}}_{{\mathcal{U}}_0}\oplus \mathbb{R}u$ and$$\begin{array}{cc}\hfill T_u{\mathcal{S}}_{{\mathcal{U}}_0}& =\{v\in {\mathbb{E}}_{{\mathcal{U}}_0}:{\int }_{{\mathbb{R}}^N}{((|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u+{\mathcal{U}}_0{|\mathrm{\nabla }u|}^{q-2}\mathrm{\nabla }u)\mathrm{\nabla }v\hfill \\ \hfill & +V_0{(|u|}^{p-2}u+{\mathcal{U}}_0{|u|}^{q-2}u)v)\text{d}x=0\}.\hfill \end{array}$$

Lemma 4.3

There holds that

1. for any $u\in {\mathbb{E}}_{{\mathcal{U}}_0}^{+},$ let $h_u:[0,\infty )\to \mathbb{R}$ be defined as $h_u(t):={\mathcal{J}}_{{\mathcal{U}}_0}(tu).$ We have that there is a unique number $t_u>0$ fulfilling $h_u^{\prime }(t)>0$ in $(0,t_u),$ and $h_u^{\prime }(t)<0$ in $(t_u,+\infty );$

2. there is $\tau >0,$ such that $t_u\ge \tau$ for every $u\in {\mathcal{S}}_{{\mathcal{U}}_0}.$ Furthermore,  if $\mathcal{K}\subset {\mathcal{S}}_{{\mathcal{U}}_0}$ is a compact set,  then there exists $C_{\mathcal{K}}>0$ only dependent on $\mathcal{K}$ such that $t_u\le C_{\mathcal{K}}$ for each $u\in \mathcal{K};$

3. set ${\widehat{m}}_{{\mathcal{U}}_0}:{\mathbb{E}}_{{\mathcal{U}}_0}^{+}\to {\mathcal{M}}_{{\mathcal{U}}_0}$ be defined as ${\widehat{m}}_{{\mathcal{U}}_0}(u):=t_{u}u.$ Then ${\widehat{m}}_{{\mathcal{U}}_0}$ is continuous and $m_{{\mathcal{U}}_0}:={\widehat{m}}_{{\mathcal{U}}_0}{|}_{{\mathcal{S}}_{{\mathcal{U}}_0}}$ is a homeomorphism between ${\mathcal{S}}_{{\mathcal{U}}_0}$ and ${\mathcal{M}}_{{\mathcal{U}}_0}.$ Moreover,  $m_{{\mathcal{U}}_0}^{-1}(u)=\frac{u}{{\Vert u\Vert }_{{\mathcal{U}}_0}};$

4. suppose that $\{u_n\}\subset {\mathcal{S}}_{{\mathcal{U}}_0}$ is a sequence such that $\mathrm{dist}({\mathrm{u}}_{\mathrm{n}},\partial {\mathcal{S}}_{{\mathcal{U}}_0})\to 0.$ Then $\Vert m_{{\mathcal{U}}_0}(u_n){\Vert }_{{\mathcal{U}}_0}\to \infty$ and ${\mathcal{J}}_{{\mathcal{U}}_0}(m_{{\mathcal{U}}_0}(u_n))\to \infty .$

Define the map ${\widehat{\phi }}_{{\mathcal{U}}_0}:{\mathbb{E}}_{{\mathcal{U}}_0}\to \mathbb{R}$ by$$\begin{array}{c}\hfill {\widehat{\phi }}_{{\mathcal{U}}_0}={\mathcal{J}}_{{\mathcal{U}}_0}({\widehat{m}}_{{\mathcal{U}}_0}(u)).\end{array}$$Let ${\phi }_{{\mathcal{U}}_0}:{\mathcal{S}}_{{\mathcal{U}}_0}\to \mathbb{R}$ be given as$$\begin{array}{c}\hfill {\phi }_{{\mathcal{U}}_0}={\widehat{\phi }}_{{\mathcal{U}}_0}{|}_{{\mathcal{S}}_{{\mathcal{U}}_0}}.\end{array}$$

Proposition 4.1

Let $(A_1)$–$(A_4)$ and $(f_1)$–$(f_4)$ hold. Then

1. ${\widehat{\phi }}_{{\mathcal{U}}_0}\in C^1({\mathbb{E}}_{{\mathcal{U}}_0}^{+},\mathbb{R})$ and $$\begin{array}{cc}\hfill ⟨{\widehat{\phi }}_{{\mathcal{U}}_0}^{\prime }(u),v⟩=& \frac{\Vert {\widehat{m}}_{{\mathcal{U}}_0}{(u)\Vert }_{{\mathcal{U}}_0}}{{\Vert u\Vert }_{{\mathcal{U}}_0}}⟨{\mathcal{J}}_{{\mathcal{U}}_0}^{\prime }({\widehat{m}}_{{\mathcal{U}}_0}(u)),v⟩\hfill \\ \hfill & \mathrm{for}\mathrm{every}\mathrm{u}\in {\mathbb{E}}_{{\mathcal{U}}_0}^{+}\mathrm{and}\mathrm{v}\in {\mathbb{E}}_{{\mathcal{U}}_0};\hfill \end{array}$$

2. ${\phi }_{{\mathcal{U}}_0}\in C^1({\mathcal{S}}_{{\mathcal{U}}_0},\mathbb{R})$ and $⟨{\phi }_{{\mathcal{U}}_0}^{\prime }(u),v⟩={\Vert m_{{\mathcal{U}}_0}(u)\Vert }_{{\mathcal{U}}_0}⟨{\mathcal{J}}_{{\mathcal{U}}_0}^{\prime }(m_{{\mathcal{U}}_0}(u)),v⟩$ for every $v\in T_u{\mathcal{S}}_{{\mathcal{U}}_0};$

3. assume that $\{u_n\}$ is a ${(PS)}_d$ sequence to ${\phi }_{{\mathcal{U}}_0}$. We have that $\{m_{{\mathcal{U}}_0}(u_n)\}$ is a ${(PS)}_d$ sequence to ${\mathcal{J}}_{{\mathcal{U}}_0}.$ Moreover,  assume that $\{u_n\}\subset {\mathcal{M}}_{{\mathcal{U}}_0}$ is a bounded ${(PS)}_d$ sequence for ${\mathcal{J}}_{{\mathcal{U}}_0}.$ We have that $\{m_{{\mathcal{U}}_0}^{-1}(u_n)\}$ is a ${(PS)}_d$ sequence for ${\phi }_{{\mathcal{U}}_0};$

4. u is a critical point of ${\phi }_{{\mathcal{U}}_0}$ if and only if $m_{{\mathcal{U}}_0}(u)$ is a nontrivial critical point of ${\mathcal{J}}_{{\mathcal{U}}_0}.$ Moreover,  the corresponding critical value coincides and $$\begin{array}{c}\hfill \underset{u\in {\mathcal{S}}_{{\mathcal{U}}_0}}{inf}{\phi }_{{\mathcal{U}}_0}(u)=\underset{u\in {\mathcal{M}}_{{\mathcal{U}}_0}}{inf}{\mathcal{J}}_{{\mathcal{U}}_0}(u).\end{array}$$

Define the set$$\begin{array}{c}\hfill \mathrm{\Gamma }=\{\gamma \in C([0,1],{\mathbb{E}}_{{\mathcal{U}}_0}):\gamma (0)=0\mathrm{and}{\mathcal{J}}_{{\mathcal{U}}_0}(\gamma (1))<0\}.\end{array}$$Then by the above arguments we can derive that

4.2

$$\begin{array}{cc}\hfill d_{{\mathcal{U}}_0}:=\underset{u\in {\mathcal{M}}_{{\mathcal{U}}_0}}{inf}{\mathcal{J}}_{{\mathcal{U}}_0}(u)& =\underset{u\in {\mathbb{E}}_{{\mathcal{U}}_0}^{+}}{inf}\underset{t\ge 0}{max}{\mathcal{J}}_{{\mathcal{U}}_0}(tu)\hfill \\ \hfill & =\underset{u\in {\mathcal{S}}_{{\mathcal{U}}_0}}{inf}\underset{t\ge 0}{max}{\mathcal{J}}_{{\mathcal{U}}_0}(tu)=\underset{\gamma \in \mathrm{\Gamma }}{inf}\underset{t\in [0,1]}{max}{\mathcal{J}}_{{\mathcal{U}}_0}(\gamma (t)).\hfill \end{array}$$

Lemma 4.4

Suppose that $\{u_n\}$ is a ${(PS)}_{d_{{\mathcal{U}}_0}}$ sequence. Then there exist $R,\alpha >0$ and $\{x_n\}\subset {\mathbb{R}}^N$ such that$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; inf}{\int }_{B_R(x_n)}{|u_n|}^p\text{d}x\ge \alpha .\end{array}$$

Proof

In contrary, we suppose that for any $R>0,$$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\underset{x\in {\mathbb{R}}^N}{sup}{\int }_{B_R(x)}{|u_n|}^p\text{d}x=0.\end{array}$$Then we can use Lemma 2.2 to infer that$$\begin{array}{c}\hfill u_n\to 0\mathrm{in}{\mathrm{L}}^{\mathrm{r}}({\mathbb{R}}^{\mathrm{N}})\mathrm{for}\mathrm{any}\mathrm{r}\in (\mathrm{p},{\mathrm{p}}^{\ast }).\end{array}$$By $(f_1),$$(f_2)$ and the Hardy–Littlewood–Sobolev inequality, one deduces that$$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast F(u_n))f(u_n)u_n\text{d}x=o_n(1).\end{array}$$Since $\{u_n\}$ is a ${(PS)}_{d_{{\mathcal{U}}_0}}$ sequence of ${\mathcal{J}}_{{\mathcal{U}}_0},$ from $(f_3)$ we conclude that $\{u_n\}$ is bounded in ${\mathbb{E}}_{{\mathcal{U}}_0}.$ Thus, we have $⟨{\mathcal{J}}_{{\mathcal{U}}_0}^{\prime }(u_n),u_n⟩=o_n(1).$ Then there holds that$$\begin{array}{c}\hfill \Vert u_n{\Vert }_p^p+{\Vert u_n\Vert }_{q,{\mathcal{U}}_0}^q={\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast F(u_n))f(u_n)u_n\text{d}x+o_n(1)=o_n(1).\end{array}$$Using this fact, we have that $u_n\to 0$ in ${\mathbb{E}}_{{\mathcal{U}}_0},$ which leads to that$$\begin{array}{c}\hfill {\mathcal{J}}_{{\mathcal{U}}_0}(u_n)\to 0.\end{array}$$This is false since$$\begin{array}{c}\hfill {\mathcal{J}}_{{\mathcal{U}}_0}(u_n)\to d_{{\mathcal{U}}_0}\mathrm{with}{\mathrm{d}}_{{\mathcal{U}}_0}>0.\end{array}$$$\square$

Remark 4.1

Let $\{u_n\}\subset {\mathbb{E}}_{{\mathcal{U}}_0}$ be a ${(PS)}_{d_{{\mathcal{U}}_0}}$ of ${\mathcal{J}}_{{\mathcal{U}}_0}.$ Then there exists $u\in {\mathbb{E}}_{{\mathcal{U}}_0}$ such that $u_n\rightharpoonup u$ in ${\mathbb{E}}_{{\mathcal{U}}_0}.$ By Lemma 4.4, we may assume that $u\ne 0$ in ${\mathbb{E}}_{{\mathcal{U}}_0}.$

Lemma 4.5

Problem (1.1) admits at least a positive ground state solution.

Proof

By the mountain pass theorem in [25], Lemmas 4.1 and 4.2, we have that there exists a sequence $\{u_n\}\subset {\mathbb{E}}_{{\mathcal{U}}_0}$ such that$$\begin{array}{c}\hfill {\mathcal{J}}_{{\mathcal{U}}_0}(u_n)\to d_{{\mathcal{U}}_0}\mathrm{and}{\mathcal{J}}_{{\mathcal{U}}_0}^{\prime }({\mathrm{u}}_{\mathrm{n}})\to 0.\end{array}$$Thereby, we can apply Remark 4.1 to derive that$$\begin{array}{c}\hfill u_n\rightharpoonup u\mathrm{in}{\mathbb{E}}_{{\mathcal{U}}_0}.\end{array}$$It is standard to conclude that ${\mathcal{J}}_{{\mathcal{U}}_0}^{\prime }(u)=0.$ We omit it here. According to $(f_3)$ and Fatou’s lemma, one holds that$$\begin{array}{cc}\hfill d_{{\mathcal{U}}_0}& =\underset{n\to \infty }{lim\; inf}[{\mathcal{J}}_{{\mathcal{U}}_0}(u_n)-\frac{1}{2p}⟨{\mathcal{J}}_{{\mathcal{U}}_0}^{\prime }(u_n),u_n⟩]\hfill \\ \hfill & =\underset{n\to \infty }{lim\; inf}[\frac{1}{p}\Vert u_n{\Vert }_p^p+\left(\frac{1}{q},-,\frac{1}{2p}\right){\Vert u_n\Vert }_{q,{\mathcal{U}}_0}^q\hfill \\ \hfill & +\frac{1}{2p}{\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,F,(u_n)\right)(f(u_n)u_n-pF(u_n))\text{d}x]\hfill \\ \hfill & \ge \frac{1}{p}{\Vert u\Vert }_p^p+\left(\frac{1}{q},-,\frac{1}{2p}\right){\Vert u\Vert }_{q,{\mathcal{U}}_0}^q\hfill \\ \hfill & +\frac{1}{2p}{\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,F,(u)\right)(f(u)u-pF(u))\text{d}x\hfill \\ \hfill & ={\mathcal{J}}_{{\mathcal{U}}_0}(u)-\frac{1}{2p}⟨{\mathcal{J}}_{{\mathcal{U}}_0}^{\prime }(u),u⟩\ge d_{{\mathcal{U}}_0},\hfill \end{array}$$which leads to that $u_n\to u$ in ${\mathbb{X}}_{{\mathcal{U}}_{\epsilon }}.$ Then u is a ground state solution of (4.1). From the regularity result in [15], one has that $u\in C_{\mathrm{loc}}^{1,\sigma }({\mathbb{R}}^N).$ Since $f(t)=0$ for $t\le 0,$ we infer $u\ge 0$ in ${\mathbb{R}}^N.$ The maximum principle in [24] shows that $u(x)>0$ in ${\mathbb{R}}^N.$$\square$

Last, we present the following relationship between $c_{\epsilon }$ and $d_{{\mathcal{U}}_0}.$

Lemma 4.6

We have that$$\begin{array}{c}\hfill \underset{\epsilon \to 0}{lim}{c}_{\epsilon }=d_{{\mathcal{U}}_0}.\end{array}$$

Proof

Give $v_{\epsilon }(x)={\eta }_{\epsilon }(x)v(x),$ where v is a positive ground state solution of (4.1) obtained by Lemma 4.5, and define ${\eta }_{\epsilon }(x)=\eta (\epsilon x),$ and $\eta \in C_c^{\infty }({\mathbb{R}}^N)$ satisfies $0\le \eta \le 1,$$\eta =1$ in $B_{\rho }(0)$ and $\eta =0$ in $B_{2\rho }(0)$ with $B_{2\rho }\subset \mathrm{\Lambda }.$ We can invoke the dominated convergence theorem to see that

4.3

$$\begin{array}{c}\hfill w_{\epsilon }\to w\mathrm{in}{\mathbb{E}}_{{\mathcal{U}}_0}\mathrm{and}{\mathcal{J}}_{{\mathcal{U}}_0}({\mathrm{w}}_{\epsilon })\to {\mathcal{J}}_{{\mathcal{U}}_0}(\mathrm{w})\mathrm{as}\epsilon \to 0.\end{array}$$For any $\epsilon >0,$ there exists $t_{\epsilon }>0$ such that$$\begin{array}{c}\hfill {\mathcal{L}}_{\epsilon }(t_{\epsilon }{w}_{\epsilon })=\underset{t\ge 0}{max}{\mathcal{L}}_{\epsilon }(t{w}_{\epsilon }).\end{array}$$Then there has that $⟨{\mathcal{L}}_{\epsilon }^{\prime }(t_{\epsilon }{w}_{\epsilon }),w_{\epsilon }⟩=0$ and

4.4

$$\begin{array}{c}\hfill t_{\epsilon }^p\Vert w_{\epsilon }{\Vert }_{p,\epsilon }^p+t_{\epsilon }^q{\Vert w_{\epsilon }\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q={\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast F(t_{\epsilon }{w}_{\epsilon }))F(t_{\epsilon }{w}_{\epsilon })\text{d}x.\end{array}$$We shall prove that $t_{\epsilon }\to 1$ as $\epsilon \to 0.$ First, we show the boundedness of $\{t_{\epsilon }\}.$ Argue by contradiction that $t_{\epsilon }\to \infty$ as $\epsilon \to 0.$ Observe that$$\begin{array}{c}\hfill t_{\epsilon }^{-p}\Vert w_{\epsilon }{\Vert }_{p,\epsilon }^p+t_{\epsilon }^{q-2p}{\Vert w_{\epsilon }\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q={\iint }_{{\mathbb{R}}^{2N}}\frac{1}{{|x-y|}^{\mu }}\frac{F(t_{\epsilon }{w}_{\epsilon }(x))}{t_{\epsilon }^p}\frac{F(t_{\epsilon }{w}_{\epsilon }(y))}{t_{\epsilon }^p}\text{d}x\text{d}y.\end{array}$$Invoking $(f_3),$ we have$$\begin{array}{cc}& {\iint }_{{\mathbb{R}}^{2N}}\frac{1}{{|x-y|}^{\mu }}\frac{F(t_{\epsilon }{w}_{\epsilon }(x))}{t_{\epsilon }^p}\frac{F(t_{\epsilon }{w}_{\epsilon }(y))}{t_{\epsilon }^p}\text{d}x\text{d}y\hfill \\ \hfill & ={\iint }_{{\mathbb{R}}^{2N}}\frac{1}{{|x-y|}^{\mu }}\frac{F(t_{\epsilon }{w}_{\epsilon }(x))}{{(t_{\epsilon }{w}_{\epsilon }(x))}^p}\frac{F(t_{\epsilon }{w}_{\epsilon }(y))}{{(t_{\epsilon }{w}_{\epsilon }(y))}^p}{(w_{\epsilon }(x)w_{\epsilon }(y))}^p\text{d}x\text{d}y\to \infty .\hfill \end{array}$$However, since $q<2p,$ we conclude that$$\begin{array}{c}\hfill t_{\epsilon }^{-p}\Vert w_{\epsilon }{\Vert }_{p,\epsilon }^p+t_{\epsilon }^{q-2p}{\Vert w_{\epsilon }\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q\to 0.\end{array}$$Then we obtain a contradiction. Thus $\{t_{\epsilon }\}$ is bounded and there exists $t_0\ge 0$ such that $t_{\epsilon }\to t_0.$ Now, we show that $t_0>0.$ Suppose by contradiction that $t_0=0.$ Thanks to $(f_1),$$(f_2),$ Lemmas 2.3 and 2.5, we deduce that$$\begin{array}{c}\hfill t_{\epsilon }^{-p}\Vert w_{\epsilon }{\Vert }_{p,\epsilon }^p+t_{\epsilon }^{q-2p}\Vert w_{\epsilon }{\Vert }_{q,{\mathcal{U}}_{\epsilon }}^q\le C(\Vert w_{\epsilon }{\Vert }_{\epsilon }^{2p}+t_{\epsilon }^{2(r_0-p)}\Vert w_{\epsilon }{\Vert }_{\epsilon }^{2r_0}).\end{array}$$As $\epsilon \to 0,$ we derive that $\infty \le C.$ This is a contradiction. Then $t_0>0.$ Let $\epsilon \to 0$ in (4.4). One has that$$\begin{array}{c}\hfill t_0^p{\Vert w\Vert }_p^p+t_0^q{\Vert w\Vert }_{q,{\mathcal{U}}_0}^q={\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast F(t_{0}w))F(t_{0}w)\text{d}x.\end{array}$$Then $t_{0}w\in {\mathcal{M}}_{{\mathcal{U}}_0}.$ Due to $w\in {\mathcal{M}}_{{\mathcal{U}}_0},$ we see $t_0=1.$ Consequently,$$\begin{array}{c}\hfill \underset{\epsilon \to 0}{lim\; sup}{c}_{\epsilon }\le \underset{\epsilon \to 0}{lim\; sup}\underset{t\ge 0}{max}{\mathcal{L}}_{\epsilon }(t{w}_{\epsilon })\le \underset{\epsilon \to 0}{lim\; sup}{\mathcal{L}}_{\epsilon }(t_{\epsilon }{w}_{\epsilon })={\mathcal{J}}_{{\mathcal{U}}_0}(t_{0}w)=c_{{\mu }_0}.\end{array}$$In view of $(A_2)$ and $(A_4),$ we have ${lim}_{\epsilon \to 0}{c}_{\epsilon }\ge d_{{\mathcal{U}}_0}.$ Then$$\begin{array}{c}\hfill \underset{\epsilon \to 0}{lim}{c}_{\epsilon }=d_{{\mathcal{U}}_0}.\end{array}$$$\square$

We provide a very important compactness result to problem (3.1).

Lemma 4.7

Assume that $\{u_n\}\subset {\mathcal{M}}_{{\mathcal{U}}_0}$ is such that ${\mathcal{J}}_{{\mathcal{U}}_0}(u_n)\to d_{{\mathcal{U}}_0}.$ Then up to a subsequence, $\{u_n\}$ has a convergent subsequence in ${\mathbb{E}}_{{\mathcal{U}}_0}.$

Proof

Let$$\begin{array}{c}\hfill v_n=m_{{\mathcal{U}}_0}^{-1}(u_n)=\frac{u_n}{\Vert u_n{\Vert }_{{\mathcal{U}}_0}}\in {\mathcal{S}}_{{\mathcal{U}}_0}.\end{array}$$Then it follows that$$\begin{array}{c}\hfill {\phi }_{{\mathcal{U}}_0}(v_n)={\mathcal{J}}_{{\mathcal{U}}_0}(u_n)\to d_{{\mathcal{U}}_0}.\end{array}$$In the light of the Ekeland variational principle, there exists $\{{\widehat{v}}_n\}\subset {\mathcal{S}}_{\epsilon }$ such that$$\begin{array}{c}\hfill {\phi }_{{\mathcal{U}}_0}({\widehat{v}}_n)\to d_{{\mathcal{U}}_0},{\phi }_{{\mathcal{U}}_0}^{\prime }({\widehat{v}}_n)\to 0\mathrm{and}{\Vert {\mathrm{v}}_{\mathrm{n}}-{\widehat{\mathrm{v}}}_{\mathrm{n}}\Vert }_{{\mathcal{U}}_0}\to 0.\end{array}$$Let ${\widehat{u}}_n=m_{{\mathcal{U}}_0}(\widehat{v_n}).$ We can derive from Proposition 4.1-(iii) that $\{{\widehat{u}}_n\}$ is a ${(PS)}_{d_{{\mathcal{U}}_0}}$ sequence of ${\mathcal{J}}_{{\mathcal{U}}_0}.$ Arguing as the proof of Lemma 4.5, we can find $\widehat{v}\in {\mathcal{S}}_{\epsilon }$ such that ${\widehat{u}}_n\to m_{{\mathcal{U}}_0}(\widehat{v}).$ Thus there holds that ${\widehat{v}}_n\to \widehat{v}.$ Then we can see that$$\begin{array}{c}\hfill v_n\to \widehat{v}\mathrm{and}{\mathrm{u}}_{\mathrm{n}}\to {\mathrm{m}}_{{\mathcal{U}}_0}(\widehat{\mathrm{v}}).\end{array}$$$\square$

The multiplicity property for (3.1)

We consider the multiplicity of solutions for the modified problem (3.1) in this section.

Let $\delta >0$ be such that

5.1

$$\begin{array}{c}\hfill M_{\delta }=\{x\in {\mathbb{R}}^N:\mathrm{dist}(\mathrm{x},\mathrm{M})\le \delta \}\subset \mathrm{\Lambda },\end{array}$$and let the function $\pi \in C^{\infty }([0,\infty ),[0,1])$ satisfy $\pi$ is nonincreasing, $\pi (t)=1$ for $t\in [0,\frac{\delta }{2}],$$\pi (t)=0$ for $t\in [\delta ,\infty )$ and $|{\pi }^{\prime }(t)|\le \frac{4}{\delta }.$ For any $y\in M,$ we define the function$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_{\epsilon ,y}(x)=\pi (|\epsilon x-y|)w(\frac{\epsilon x-y}{\epsilon }),\end{array}$$where w is positive ground state solution of the autonomous problem (4.1).

Take the map ${\mathrm{{\rm Y}}}_{\epsilon }:M\to {\mathcal{N}}_{\epsilon }$ defined as$$\begin{array}{c}\hfill {\mathrm{{\rm Y}}}_{\epsilon }=t_{\epsilon }{\mathrm{\Gamma }}_{\epsilon ,y},\end{array}$$where $t_{\epsilon }$ is defined by$$\begin{array}{c}\hfill {\mathcal{L}}_{\epsilon }(t_{\epsilon }{\mathrm{\Gamma }}_{\epsilon ,y})=\underset{t\ge 0}{max}{\mathcal{L}}_{\epsilon }(t{\mathrm{\Gamma }}_{\epsilon ,y}).\end{array}$$

Lemma 5.1

The map ${\mathrm{{\rm Y}}}_{\epsilon }$ satisfies the relation$$\begin{array}{c}\hfill \underset{\epsilon \to \infty }{lim}{\mathcal{L}}_{\epsilon }({\mathrm{{\rm Y}}}_{\epsilon }(y))=d_{{\mathcal{U}}_0}\mathrm{uniformly}\mathrm{in}\mathrm{y}\in \mathrm{M}.\end{array}$$

Proof

Arguing by contradiction, we can find ${\epsilon }_n\to 0,$$\{y_n\}\subset M$ and ${\beta }_0>0$ fulfilling

5.2

$$\begin{array}{c}\hfill |{\mathcal{L}}_{{\epsilon }_n}({\mathrm{{\rm Y}}}_{{\epsilon }_n}(y_n))-d_{{\mathcal{U}}_0}|\ge {\beta }_0\mathrm{for}\mathrm{any}\mathrm{n}\in \mathbb{N}.\end{array}$$From the fact $⟨{\mathcal{L}}_{{\epsilon }_n}^{\prime }({\mathrm{{\rm Y}}}_{{\epsilon }_n}(y_n)),{\mathrm{{\rm Y}}}_{{\epsilon }_n}(y_n)⟩=0,$ we get that

5.3

$$\begin{array}{cc}& t_{{\epsilon }_n}^p\Vert {\mathrm{\Gamma }}_{{\epsilon }_n,y_n}{\Vert }_{p,{\epsilon }_n}^p+t_{{\epsilon }_n}^q{\Vert {\mathrm{\Gamma }}_{{\epsilon }_n,y_n}\Vert }_{q,{\mu }_{{\epsilon }_n}}^q\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{{\epsilon }_n},(x,t_{{\epsilon }_n}{\mathrm{\Gamma }}_{{\epsilon }_n,y_n}(x))\right)g_{{\epsilon }_n}(x,t_{{\epsilon }_n}{\mathrm{\Gamma }}_{{\epsilon }_n,y_n}(x))t_{{\epsilon }_n}{\mathrm{\Gamma }}_{{\epsilon }_n,y_n}(x)\text{d}x.\hfill \end{array}$$By the change of variable $z=\frac{{\epsilon }_{n}x-y_n}{{\epsilon }_n},$ we have that$$\begin{array}{c}\hfill \Vert {\mathrm{\Gamma }}_{{\epsilon }_n,y_n}{\Vert }_{p,{\epsilon }_n}^p={\int }_{{\mathbb{R}}^N}(|\mathrm{\nabla }(\pi (|{\epsilon }_n{z|)e(z))|}^p+V({\epsilon }_{n}z+y_n)|\pi (|{\epsilon }_{n}z|)w(z){|}^p)dz.\end{array}$$Let ${\epsilon }_n\to 0$ in the above formula. Using the Lebesgue’s dominated convergence theorem, there holds

5.4

$$\begin{array}{c}\hfill \Vert {\mathrm{\Gamma }}_{{\epsilon }_n,y_n}{\Vert }_{p,{\epsilon }_n}^p\to {\int }_{{\mathbb{R}}^N}{(|\mathrm{\nabla }w|}^p+V_0{|w|}^p)\text{d}x.\end{array}$$By a similar way, we have

5.5

$$\begin{array}{c}\hfill \Vert {\mathrm{\Gamma }}_{{\epsilon }_n,y_n}{\Vert }_{q,{\mu }_{{\epsilon }_n}}^q\to {\mathcal{U}}_0{\Vert w\Vert }_q^q.\end{array}$$Again by the changes of variable $x^{\prime }=\frac{{\epsilon }_{n}x-y_n}{{\epsilon }_n},$$y^{\prime }=\frac{{\epsilon }_{n}y-y_n}{{\epsilon }_n},$${\epsilon }_n{x}^{\prime }+y_n\in M_{\delta }\subset \mathrm{\Lambda }$ for any $|{\epsilon }_n{x}^{\prime }|<\delta$ and ${\epsilon }_n{y}^{\prime }+y_n\in M_{\delta }\subset \mathrm{\Lambda }$ for any $|{\epsilon }_n{y}^{\prime }|<\delta$, we conclude that

5.6

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast G_{{\epsilon }_n}(t_{{\epsilon }_n}{\mathrm{\Gamma }}_{{\epsilon }_n,y_n}(x)))g_{{\epsilon }_n}(x,t_{{\epsilon }_n}{\mathrm{\Gamma }}_{{\epsilon }_n,y_n}(x))t_{{\epsilon }_n}{\mathrm{\Gamma }}_{{\epsilon }_n,y_n}(x)\text{d}x\hfill \\ \hfill & ={\iint }_{{\mathbb{R}}^{2N}}\frac{G({\epsilon }_n{x}^{\prime }+y_n,t_{{\epsilon }_n}\pi (|{\epsilon }_n{x}^{\prime }|)w(x^{\prime }))}{|x^{\prime }-y^{\prime }{|}^{\mu }}g({\epsilon }_n{y}^{\prime }\hfill \\ \hfill & +y_n,t_{{\epsilon }_n}\pi (|{\epsilon }_n{y}^{\prime }|)w(y^{\prime }))t_{{\epsilon }_n}\pi (|{\epsilon }_n{y}^{\prime }|)w(y^{\prime })\text{d}{x}^{\prime }\text{d}{y}^{\prime }\hfill \\ \hfill & ={\iint }_{{\mathbb{R}}^{2N}}\frac{F(t_{{\epsilon }_n}\pi (|{\epsilon }_{n}x|)w(x))}{{|x-y|}^{\mu }}f(t_{{\epsilon }_n}\pi (|{\epsilon }_{n}y|)w(y))t_{{\epsilon }_n}\pi (|{\epsilon }_{n}y|)w(y)\text{d}x\text{d}y.\hfill \end{array}$$First of all, we shall prove that $\{t_{{\epsilon }_n}\}$ is bounded. By (5.3) and (5.6), we conclude that

5.7

$$\begin{array}{cc}& \frac{1}{t_{{\epsilon }_n}^{q-p}}\Vert {\mathrm{\Gamma }}_{{\epsilon }_n,y_n}{\Vert }_{p,{\epsilon }_n}^p+{\Vert {\mathrm{\Gamma }}_{{\epsilon }_n,y_n}\Vert }_{q,{\mu }_{{\epsilon }_n}}^q\hfill \end{array}$$

5.8

$$\begin{array}{cc}& ={\iint }_{{\mathbb{R}}^{2N}}\frac{1}{{|x-y|}^{\mu }}\frac{F(t_{{\epsilon }_n}\pi (|{\epsilon }_{n}x|)w(x))}{(t_{{\epsilon }_n}\pi (|{\epsilon }_n{x|)w(x))}^{\frac{q}{2}}}\frac{f(t_{{\epsilon }_n}\pi (|{\epsilon }_{n}y|)w(y))}{(t_{{\epsilon }_n}\pi (|{\epsilon }_n{y|)w(y))}^{\frac{q}{2}-1}}\hfill \\ \hfill & (\pi (|{\epsilon }_n{x|)w(x))}^{\frac{q}{2}}(\pi (|{\epsilon }_n{y|)w(y))}^{\frac{q}{2}}\text{d}x\text{d}y.\hfill \end{array}$$Notice that $\pi (|{\epsilon }_{n}z|)=1$ if $|{\epsilon }_{n}z|\le \frac{\delta }{2}$ and $\frac{f(t)}{t^{q-1}}$ is increasing in $(0,\infty ).$ We conclude from (5.7) that

5.9

$$\begin{array}{cc}& \frac{1}{t_{{\epsilon }_n}^{q-p}}\Vert {\mathrm{\Gamma }}_{{\epsilon }_n,y_n}{\Vert }_{p,{\epsilon }_n}^p+{\Vert {\mathrm{\Gamma }}_{{\epsilon }_n,y_n}\Vert }_{q,{\mu }_{{\epsilon }_n}}^q\hfill \\ \hfill & \ge \frac{F(t_{{\epsilon }_n}w(\widehat{z}))}{{(t_{{\epsilon }_n}w(\widehat{z}))}^{\frac{q}{2}}}\frac{f(t_{{\epsilon }_n}w(\widehat{z}))}{{(t_{{\epsilon }_n}w(\widehat{z}))}^{\frac{q}{2}-1}}{\iint }_{B_{\frac{\delta }{2}}\times B_{\frac{\delta }{2}}}\frac{1}{{|x-y|}^{\mu }}{(w(x))}^{\frac{q}{2}}{(w(y))}^{\frac{q}{2}}\text{d}x\text{d}y,\hfill \end{array}$$where$$\begin{array}{c}\hfill w(\widehat{z})=\underset{|z|\le \frac{\delta }{2}}{min}w(z)>0.\end{array}$$Let $n\to \infty$ in (5.9). By $(f_3),$$q<2p,$ (5.4) and (5.5), one deduces that$$\begin{array}{c}\hfill {\mathcal{U}}_0{\Vert w\Vert }_{q,{\mathcal{U}}_0}^q\ge \infty .\end{array}$$We reach a contradiction. Then we can find a number $t_0\ge 0$ such that$$\begin{array}{c}\hfill t_n\to t_0.\end{array}$$In the light of (5.2), (5.4), (5.5) and Lemma 2.5 we can reach that $t_0>0.$ Let $n\to \infty$ in (5.3). Using (5.4), (5.5) and the Lebesgue’s dominated convergence theorem, there holds that$$\begin{array}{c}\hfill t_0^p{\Vert w\Vert }_p^p+t_0^q{\Vert w\Vert }_{q,{\mathcal{U}}_0}^q={\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast F(t_{0}w))f(t_{0}w)t_{0}w\text{d}x,\end{array}$$which suggests that $t_{0}w\in {\mathcal{M}}_{{\mathcal{U}}_0}.$ Since $w\in {\mathcal{M}}_{{\mathcal{U}}_0},$ then $t_0=1$ and $t_n\to 1.$ By virtue of (5.4), (5.5), $t_n\to 1$ and Lemma 2.5, we conclude that

5.10

$$\begin{array}{cc}& {\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast G_{{\epsilon }_n}(t_{{\epsilon }_n}{\mathrm{\Gamma }}_{{\epsilon }_n,y_n}(x)))G_{{\epsilon }_n}(x,t_{{\epsilon }_n}{\mathrm{\Gamma }}_{{\epsilon }_n,y_n}(x))\text{d}x\hfill \\ \hfill & \to {\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast F(w))F(w)\text{d}x.\hfill \end{array}$$By (5.4), (5.5) and (5.10), we reach that

5.11

$$\begin{array}{c}\hfill {\mathcal{L}}_{{\epsilon }_n}({\mathrm{{\rm Y}}}_{{\epsilon }_n}(y_n))\to {\mathcal{J}}_{{\mathcal{U}}_0}(w)=d_{{\mathcal{U}}_0}.\end{array}$$Compared with (5.2) and (5.11), we obtain a contradiction. $\square$

We introduce the set$$\begin{array}{c}\hfill \widehat{{\mathcal{N}}_{\epsilon }}=\{u\in {\mathcal{N}}_{\epsilon }:{\mathcal{L}}_{\epsilon }(u)\le c_{{\mu }_0}+h_{\epsilon }\},\end{array}$$where $h_{\epsilon }:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ satisfies $h_{\epsilon }\to 0$ as $\epsilon \to 0.$

Set $h_{\epsilon }={sup}_{y\in M}|{\mathcal{L}}_{\epsilon }({\mathrm{{\rm Y}}}_{\epsilon }(y))-c_{{\mu }_0}|.$ Recalling Lemma 5.1, we can see that $h_{\epsilon }\to 0$ as $\epsilon \to 0$ and $\widehat{{\mathcal{N}}_{\epsilon }}$ is not empty.

For any $\rho >0$ being such that $M_{\delta }\subset B_{\rho },$ we define the map $\mathrm{\Sigma }:{\mathbb{R}}^N\to {\mathbb{R}}^N$ as$$\begin{array}{c}\hfill \mathrm{\Sigma }(x)=\left\{\begin{array}{cc}x\hfill & \mathrm{if}|\mathrm{x}|<\rho ,\hfill \\ \frac{\rho x}{|x|}\hfill & \mathrm{if}|\mathrm{x}|\ge \rho .\hfill \end{array}\right)\end{array}$$Let us give the barycenter map ${\mathrm{\Phi }}_{\epsilon }:{\mathcal{N}}_{\epsilon }\to {\mathbb{R}}^N$ by$$\begin{array}{c}\hfill {\mathrm{\Phi }}_e(u)=\frac{{\displaystyle {\int }_{R^N}\mathrm{\Sigma }(ex){|u(x)|}^{p}dx}}{{\displaystyle {\int }_{R^N}{|u(x)|}^{p}dx}}.\end{array}$$

Lemma 5.2

We have the following limits.$$\begin{array}{c}\hfill \underset{\epsilon \to 0}{lim}{\mathrm{\Phi }}_{\epsilon }({\mathrm{{\rm Y}}}_{\epsilon }(y))=y\mathrm{uniformly}\mathrm{in}\mathrm{y}\in \mathrm{M}.\end{array}$$

Proof

In contrary, assume that there exist ${\epsilon }_n\to 0,$$\{y_n\}\subset M$ and $\varsigma >0$ such that

5.12

$$\begin{array}{c}\hfill |{\mathrm{\Phi }}_{{\epsilon }_n}({\mathrm{{\rm Y}}}_{{\epsilon }_n}(y_n))-y_n|\ge \varsigma .\end{array}$$By changing the variable $z=\frac{{\epsilon }_{n}x-y_n}{{\epsilon }_n}$ and the fact that ${\epsilon }_{n}z+y_n\in B_{\rho }$ for any $|{\epsilon }_{n}z|<\delta$ and $y_n\in M,$ one concludes that

5.13

$$\begin{array}{cc}\hfill {\mathrm{\Phi }}_{{\epsilon }_n}({\mathrm{{\rm Y}}}_{{\epsilon }_n}(y_n))& =\frac{{\displaystyle {\int }_{{\mathbb{R}}^N}\mathrm{\Sigma }({\epsilon }_{n}x){|t_{{\epsilon }_n}{\mathrm{\Gamma }}_{{\epsilon }_n,y_n}(x)|}^p\text{d}x}}{{\displaystyle {\int }_{{\mathbb{R}}^N}{|t_{{\epsilon }_n}{\mathrm{\Gamma }}_{{\epsilon }_n,y_n}(x)|}^p\text{d}x}}\hfill \\ \hfill & =\frac{{\displaystyle {\int }_{{\mathbb{R}}^N}\mathrm{\Sigma }({\epsilon }_{n}z+y_n)|\pi (|{\epsilon }_n{z|)|}^p{|w(z)|}^p\text{d}x}}{{\displaystyle {\int }_{{\mathbb{R}}^N}|\pi (|{\epsilon }_n{z|)|}^p{|w(z)|}^p\text{d}x}}\hfill \\ \hfill & =\frac{{\displaystyle {\int }_{{\mathbb{R}}^N}({\epsilon }_{n}z+y_n)\pi (|{\epsilon }_n{z|)|}^p{|w(z)|}^p\text{d}x}}{{\displaystyle {\int }_{{\mathbb{R}}^N}|\pi (|{\epsilon }_n{z|)|}^p{|w(z)|}^p\text{d}x}}\hfill \\ \hfill & =y_n+\frac{{\displaystyle {\int }_{{\mathbb{R}}^N}{\epsilon }_{n}z|\pi (|{\epsilon }_n{z|)|}^p{|w(z)|}^p\text{d}x}}{{\displaystyle {\int }_{{\mathbb{R}}^N}|\pi (|{\epsilon }_n{z|)|}^p{|w(z)|}^p\text{d}x}}.\hfill \end{array}$$By the Lebesgue’s dominated convergence theorem, we derive that

5.14

$$\begin{array}{c}\hfill \frac{{\displaystyle {\int }_{{\mathbb{R}}^N}{\epsilon }_{n}z|\pi (|{\epsilon }_n{z|)|}^p{|w(z)|}^p\text{d}x}}{{\displaystyle {\int }_{{\mathbb{R}}^N}|\pi (|{\epsilon }_n{z|)|}^p{|w(z)|}^p\text{d}x}}\to 0.\end{array}$$It follows from (5.13) and (5.14) that

5.15

$$\begin{array}{c}\hfill |{\mathrm{\Phi }}_{{\epsilon }_n}({\mathrm{{\rm Y}}}_{{\epsilon }_n}(y_n))-y_n|\to 0.\end{array}$$(5.15) contradicts to (5.12). $\square$

The following compactness result is very useful to verify the multiplicity of solutions to (1.1).

Proposition 5.1

Suppose that ${\epsilon }_n\to 0,$$\{u_n\}\subset {\mathcal{N}}_{{\epsilon }_n}$ and ${\mathcal{L}}_{{\epsilon }_n}(u_n)\to d_{{\mathcal{U}}_0}.$ We have that there exists a sequence $\{{\widehat{y}}_n\}\subset {\mathbb{R}}^n.$ Define $v_n(x)=u_n(x+{\widehat{y}}_n).$ There holds that $\{v_{{\epsilon }_n}\}$ admits a convergent subsequence in ${\mathbb{E}}_{{\mathcal{U}}_0}.$ Besides,  there exists $y_0\in M$ such that up to a subsequence,  $y_n:={\epsilon }_n{\widehat{y}}_n\to y_0.$

Proof

Utilizing $⟨{\mathcal{L}}_{{\epsilon }_n}^{\prime }(u_n),u_n⟩=0$ and ${\mathcal{L}}_{{\epsilon }_n}(u_n)\to d_{{\mathcal{U}}_0},$ it is easy to deduce that $\{u_{{\epsilon }_n}\}$ is bounded in ${\mathbb{X}}_{{\mu }_{{\epsilon }_n}}.$ Similar to Lemma 4.4, we have that there exist $\{{\widehat{y}}_n\}\subset {\mathbb{R}}^N$ and $R,{\beta }_0>0$ such that

5.16

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; inf}{\int }_{B_R({\widehat{y}}_n)}{|u_n|}^p\text{d}x\ge {\beta }_0.\end{array}$$Let $v_n(x)=u_n(x+{\widehat{y}}_n).$ One has that $\{v_n\}$ is bounded in ${\mathbb{E}}_{{\mathcal{U}}_0}.$ Then in the sense of a subsequence, by (5.16) we can obtain that there exists $v\in {\mathbb{E}}_{{\mathcal{U}}_0}$ with $v\ne 0$ such that $v_n\rightharpoonup v$ in ${\mathbb{E}}_{{\mathcal{U}}_0}.$ Let $t_n>0$ be such that $t_n{v}_n\in {\mathcal{M}}_{{\mathcal{U}}_0}.$ Take ${\widehat{v}}_n=t_n{v}_n$ and $y_n={\epsilon }_n{\widehat{y}}_n.$ Since ${\widehat{v}}_n\in {\mathcal{M}}_{{\mathcal{U}}_0},$$\mathcal{U}(x)\ge {\mathcal{U}}_0$ for any $x\in {\mathbb{R}}^N,$$V(x)\ge V_0$ for any $x\in {\mathbb{R}}^N$ and $G(x,t)\le F(t)$ for any $x\in {\mathbb{R}}^N$ and $t\in \mathbb{R},$ we conclude that$$\begin{array}{c}\hfill d_{{\mathcal{U}}_0}\le {\mathcal{J}}_{{\mathcal{U}}_0}({\widehat{v}}_n)\le \underset{t\ge 1}{max}{\mathcal{L}}_{{\epsilon }_n}(t{v}_n)={\mathcal{L}}_{{\epsilon }_n}(v_n)=d_{{\mathcal{U}}_0}+o_n(1).\end{array}$$Thus we have that$$\begin{array}{c}\hfill {\mathcal{J}}_{{\mathcal{U}}_0}({\widehat{v}}_n)\to d_{{\mathcal{U}}_0}\mathrm{and}⟨{\mathcal{J}}_{{\mathcal{U}}_0}^{\prime }({\widehat{\mathrm{v}}}_{\mathrm{n}}),{\widehat{\mathrm{v}}}_{\mathrm{n}}⟩=0.\end{array}$$Then $\{{\widehat{v}}_n\}$ is bounded in ${\mathbb{E}}_{{\mathcal{U}}_0}$ and we may assume that ${\widehat{v}}_n\rightharpoonup \widehat{v}$ in ${\mathbb{E}}_{{\mathcal{U}}_0}.$ By a simple argument, we infer from (5.16) that there exists $t_0\in (0,\infty )$ such that $t_n\to t_0.$ This leads to $\widehat{v}\ne 0$ in ${\mathbb{E}}_{{\mathcal{U}}_0}.$ In view of Lemma 4.7, we deduce that ${\widehat{v}}_n\to \widehat{v}$ in ${\mathbb{E}}_{{\mathcal{U}}_0}.$ Moreover, we can obtain $v_n\to v$ in ${\mathbb{E}}_{{\mathcal{U}}_0}.$ So, one has$$\begin{array}{c}\hfill {\mathcal{J}}_{{\mathcal{U}}_0}(\widehat{v})=d_{{\mathcal{U}}_0}\mathrm{and}⟨{\mathcal{J}}_{{\mathcal{U}}_0}^{\prime }(\widehat{\mathrm{v}}),\widehat{\mathrm{v}}⟩=0.\end{array}$$Last we show the boundedness of $\{y_n\}.$ In contrary assume that $|y_n|\to \infty .$ Noticing that $⟨{\mathcal{L}}_{{\epsilon }_n}^{\prime }(u_n),u_n⟩=0$ and ${\mathcal{L}}_{{\epsilon }_n}(u_n)\to d_{{\mathcal{U}}_0},$ we deduce that$$\begin{array}{c}\hfill u_n\in \mathcal{K}.\end{array}$$One can derive from Lemma 3.2 that

5.17

$$\begin{array}{c}\hfill |\frac{1}{{|x|}^{\mu }}\ast F(u_n)|\le \frac{k}{2}.\end{array}$$Take $R>0$ such that $\mathrm{\Lambda }\subset B_R.$ Then we have that $|y_n|>2R$ and for any $x\in B_{\frac{R}{{\epsilon }_n}},$$$\begin{array}{c}\hfill |{\epsilon }_{n}x+y_n|\ge |y_n|-|{\epsilon }_{n}x|>R.\end{array}$$By the fact that $(A_1),$$(A_4),$$v_n\to v$ in ${\mathbb{E}}_{{\mathcal{U}}_0}$ and (5.17), we infer$$\begin{array}{cc}\hfill \Vert v_n{\Vert }_p^p+{\Vert v_n\Vert }_{q,{\mathcal{U}}_0}^q& \le \frac{k}{2}{\int }_{{\mathbb{R}}^N}g({\epsilon }_{n}x+y_n,v_n)v_n\text{d}x\hfill \\ \hfill & \le \frac{k}{2}{\int }_{B_{\frac{R}{{\epsilon }_n}}}g({\epsilon }_{n}x+y_n,v_n)v_n\text{d}x\hfill \\ \hfill & +\frac{k}{2}{\int }_{B_{\frac{R}{{\epsilon }_n}}^c}g({\epsilon }_{n}x+y_n,v_n)v_n\text{d}x\hfill \\ \hfill & \le \frac{V_0}{2}{\int }_{B_{\frac{R}{{\epsilon }_n}}}{|v_n|}^p\text{d}x+o_n(1),\hfill \end{array}$$which implies that$$\begin{array}{c}\hfill v_n\to 0\mathrm{in}{\mathbb{E}}_{{\mathcal{U}}_0}.\end{array}$$It is impossible since $v_n\to v$ in ${\mathbb{E}}_{{\mathcal{U}}_0}$ with $v\ne 0.$ Then $\{y_n\}$ is bounded and we may assume that $y_n\to y_0.$ Claim that $y_0\in \overline{\mathrm{\Lambda }}.$ In contrary we assume that $y_0\notin \overline{\mathrm{\Lambda }}.$ Then there exists ${\rho }_0>0$ such that $B_{2{\rho }_0}(y_0)\subset {\overline{\mathrm{\Lambda }}}^c.$ For any $x\in B_{\frac{{\rho }_0}{{\epsilon }_n}}(y_0),$ there holds ${\epsilon }_{n}x+y_n\in B_{2{\rho }_n}(y_0).$ Proceeding as above we can reach a contradiction. Thereby $y_0\in \overline{\mathrm{\Lambda }}.$ Assume by contradiction that $V(y_0)<V_0.$ Since $v_n\to v$ in ${\mathbb{E}}_{{\mathcal{U}}_0},$ the invariance of ${\mathbb{R}}^N$ under translation, one infers that$$\begin{array}{cc}& d_{{\mathcal{U}}_0}\le {\mathcal{J}}_{{\mathcal{U}}_0}(\widehat{v})\hfill \\ \hfill & <\underset{n\to \infty }{lim\; inf}(\frac{1}{p}|\mathrm{\nabla }{v}_n{|}_p^p+\frac{1}{p}{\int }_{{\mathbb{R}}^N}V({\epsilon }_{n}x+y_n){|v_n|}^p\text{d}x\hfill \\ \hfill & +{\mathcal{U}}_0\left(\frac{1}{q},|\mathrm{\nabla },v_n,{|}_q^q,+,\frac{1}{q},{\int }_{{\mathbb{R}}^N},V,({\epsilon }_{n}x+y_n),{|v_n|}^q,\text{d},x\right)\hfill \\ \hfill & -{\int }_{{\mathbb{R}}^N}\left(\frac{1}{{|x|}^{\mu }},,\ast ,F,(v_n)\right)F(v_n)\text{d}x)\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; inf}{\mathcal{L}}_{{\epsilon }_n}(t_n{u}_n)=\underset{n\to \infty }{lim\; inf}{\mathcal{L}}_{{\epsilon }_n}(u_n)=d_{{\mathcal{U}}_0}.\hfill \end{array}$$This is false. Then $y_0\in M$ and $y_n\to y_0.$$\square$

Lemma 5.3

Let $\delta >0$ be such that $M_{\delta }\subset \mathrm{\Lambda }.$ We have$$\begin{array}{c}\hfill \underset{\epsilon \to 0}{lim}\underset{u\in \widehat{{\mathcal{N}}_{\epsilon }}}{sup}\mathrm{dist}({\mathrm{\Phi }}_{\epsilon }(\mathrm{u}),{\mathrm{M}}_{\delta })=0.\end{array}$$

Proof

Take ${\epsilon }_n\to 0,$ then there exists $\{u_n\}\subset {\mathcal{N}}_{{\epsilon }_n}$ such that$$\begin{array}{c}\hfill \underset{u\in {\widehat{\mathcal{N}}}_{{\epsilon }_n}}{sup}\underset{x\in M_{\delta }}{inf}|{\mathrm{\Phi }}_{{\epsilon }_n}(u)-x|=\underset{x\in M_{\delta }}{inf}|{\mathrm{\Phi }}_{{\epsilon }_n}(u_n)-x|+o_n(1).\end{array}$$It is enough to find $\{x_n\}\subset M_{\delta }$ such that$$\begin{array}{c}\hfill |{\mathrm{\Phi }}_{{\epsilon }_n}(u_n)-x_n|\to 0.\end{array}$$From $\{u_n\}\subset {\widehat{\mathcal{N}}}_{{\epsilon }_n}\subset {\mathcal{N}}_{{\epsilon }_n}$ and ${\mathcal{J}}_{\epsilon }(t{u}_n)\le {\mathcal{L}}_{\epsilon }(t{u}_n)$ for any $t\ge 0,$ we deduce$$\begin{array}{c}\hfill d_{{\mathcal{U}}_0}\le \underset{t\ge 0}{max}{\mathcal{J}}_{{\mathcal{U}}_0}(t{u}_n)\le \underset{t\ge 0}{max}{\mathcal{L}}_{{\epsilon }_n}(t{u}_n)\le {\mathcal{L}}_{{\epsilon }_n}(u_n)\le d_{{\mathcal{U}}_0}+h_{{\epsilon }_n}.\end{array}$$This leads to ${\mathcal{L}}_{{\epsilon }_n}(u_n)\to d_{{\mathcal{U}}_0}.$ By virtue of Proposition 5.1, there exist ${\widehat{x}}_n\in {\mathbb{R}}^N$ and $x_n={\epsilon }_n{\widehat{x}}_n$ such that $x_n\in M_{\delta }.$ We calculate that$$\begin{array}{c}\hfill {\mathrm{\Phi }}_{{\epsilon }_n}(u_n)=x_n+\frac{{\int }_{{\mathbb{R}}^N}(\mathrm{\Sigma }({\epsilon }_{n}x+x_n)-x_n){|u_n(x+{\widehat{x}}_n)|}^p\text{d}x}{{\int }_{{\mathbb{R}}^N}{|u_n(x+{\widehat{x}}_n)|}^p\text{d}x}.\end{array}$$It follows from Proposition 5.1 that ${\epsilon }_{n}z+x_n\to x_0\in M_{\delta }$ for any $z\in {\mathbb{R}}^N$ and $\{u_n(·+{\widehat{x}}_n)\}$ is convergent in ${\mathbb{E}}_{{\mathcal{U}}_0}.$ Then we can use the dominated convergence theorem to see that$$\begin{array}{c}\hfill {\mathrm{\Phi }}_{{\epsilon }_n}(u_n)-x_n=o_n(1).\end{array}$$$\square$

Lemma 5.4

Assume that $(A_1)$–$(A_4)$ and $(f_1)$–$(f_4)$ hold. Then for any $\delta >0$ being such that $M_{\delta }\subset \mathrm{\Lambda },$ there exists ${\epsilon }_{\delta }>0$ such that for any $\epsilon \in (0,{\epsilon }_{\delta }),$ problem (3.1) admits at least ${\mathrm{cat}}_{{\mathrm{M}}_{\delta }}(\mathrm{M})$ nonnegative solutions.

Proof

Let us define the set ${\widehat{\mathcal{S}}}_{\epsilon }$ by$$\begin{array}{c}\hfill {\widehat{\mathcal{S}}}_{\epsilon }=\{u\in {\mathcal{S}}_{\epsilon }:{\phi }_{\epsilon }(u)\le d_{{\mathcal{U}}_0}+h_{\epsilon }\}.\end{array}$$We can observe that for any $u\in {\widehat{\mathcal{N}}}_{\epsilon },$$$\begin{array}{c}\hfill {\phi }_{\epsilon }(m_{\epsilon }^{-1}(u))=({\mathcal{L}}_{\epsilon }\circ m_{\epsilon })(m_{\epsilon }^{-1}(u))={\mathcal{L}}_{\epsilon }(u)\le d_{{\mathcal{U}}_0}+h_{\epsilon },\end{array}$$from which, we infer that the map $m_{\epsilon }^{-1}:{\widehat{\mathcal{N}}}_{\epsilon }\to {\widehat{\mathcal{S}}}_{\epsilon }$ is well defined. Similarly, we can derive that the map $m_{\epsilon }:{\widehat{\mathcal{S}}}_{\epsilon }\to {\widehat{\mathcal{N}}}_{\epsilon }$ is well defined. Exploiting Lemmas 5.1 and 5.3, there exists ${\epsilon }_{\delta }^1>0$ such that for any $\epsilon \in (0,{\epsilon }_{\delta }^1),$ we have the following well-defined diagram$$\begin{array}{c}\hfill M\stackrel{{\mathrm{{\rm Y}}}_{\epsilon }}{⟶}{\widehat{\mathcal{N}}}_{\epsilon }\stackrel{m_{\epsilon }^{-1}}{⟶}{\widehat{\mathcal{S}}}_{\epsilon }\stackrel{m_{\epsilon }}{⟶}{\widehat{\mathcal{N}}}_{\epsilon }\stackrel{{\mathrm{\Phi }}_{\epsilon }}{⟶}{M}_{\delta }.\end{array}$$Then we can derive the following well-defined diagram$$\begin{array}{c}\hfill M\stackrel{m_{\epsilon }^{-1}\circ {\mathrm{{\rm Y}}}_{\epsilon }}{⟶}{\widehat{\mathcal{S}}}_{\epsilon }\stackrel{{\mathrm{\Phi }}_{\epsilon }\circ m_{\epsilon }}{⟶}{M}_{\delta }.\end{array}$$From Lemma 5.2, there exists ${\epsilon }_{\delta }^2>0$ such that for any $\epsilon \in (0,{\epsilon }_{\delta }^2),$$$\begin{array}{c}\hfill ({\mathrm{\Phi }}_{\epsilon }\circ m_{\epsilon })(m_{\epsilon }^{-1}\circ {\mathrm{{\rm Y}}}_{\epsilon })(y)=({\mathrm{\Phi }}_{\epsilon }{\mathrm{{\rm Y}}}_{\epsilon })(y)=y+\varphi (\epsilon ,y)\mathrm{for}\mathrm{each}\mathrm{y}\in \mathrm{M},\end{array}$$where $|\varphi (\epsilon ,y)|\le \frac{\delta }{2}$ for each $\epsilon \in (0,{\epsilon }_{\delta }^2)$ and $y\in M.$ Define the map $H:[0,1]\times M\to M_{\delta }$ by $H(t,y)=y+t\varphi (\epsilon ,y)$ for each $(t,y)\in [0,1]\times M.$ It is easy to see that H is a homotopy equivalent map between the including map $\mathrm{id}$ and the map $({\mathrm{\Phi }}_{\epsilon }\circ m_{\epsilon })(m_{\epsilon }^{-1}\circ {\mathrm{{\rm Y}}}_{\epsilon }).$ According to [5, Theorem 6.3.21], one has$$\begin{array}{c}\hfill {\mathrm{cat}}_{{\widehat{\mathcal{S}}}_{\epsilon }}({\widehat{\mathcal{S}}}_{\epsilon })\ge {\mathrm{cat}}_{{\mathrm{M}}_{\delta }}(\mathrm{M}).\end{array}$$Invoking Lemmas 3.2, and 4.6, there exists ${\epsilon }_{\delta }^3>0$ such that for any $\epsilon \in (0,{\epsilon }_{\delta }^3),$${\phi }_{\epsilon }$ satisfies ${(PS)}_c$ condition on ${\widehat{\mathcal{S}}}_{\epsilon }$ for $c_{\epsilon }\le c\le \mathcal{B}.$ Using [5, Theorem 6.3.20], ${\phi }_{\epsilon }$ has at least ${\mathrm{cat}}_{{\widehat{\mathcal{S}}}_{\epsilon }}({\widehat{\mathcal{S}}}_{\epsilon })$ critical points. Let ${\epsilon }_{\delta }=min\{{\epsilon }_{\delta }^1,{\epsilon }_{\delta }^2,{\epsilon }_{\delta }^3\}.$ By Proposition 4.1, ${\mathcal{L}}_{\epsilon }$ has at least ${\mathrm{cat}}_{{\mathrm{M}}_{\delta }}(\mathrm{M})$ critical points. Moreover, we can deduce that every critical point of ${\mathcal{L}}_{\epsilon }$ is nonnegative. $\square$

Proof of Theorem 1.1

We start with the $L^{\infty }$ estimates and decaying estimates of solutions for problem (3.1), which play important roles in verifying the concentrating phenomenon.

Lemma 6.1

Let ${\epsilon }_n\to 0$ and $u_n\in {\widehat{\mathcal{N}}}_{{\epsilon }_n}$ be a solution of Eq. (3.1). Set $v_n=u_n(·+{\widehat{y}}_n)$ be given by Proposition 5.1. Then there exists a constant $C>0$ independent of n such that$$\begin{array}{c}\hfill |v_n{|}_{\infty }\le C\mathrm{uniformly}\mathrm{for}\mathrm{any}\mathrm{n}\in \mathbb{N}.\end{array}$$Besides, $$\begin{array}{c}\hfill \underset{|x|\to \infty }{lim}{v}_n(x)=0\mathrm{uniformly}\mathrm{for}\mathrm{n}\in \mathbb{N}.\end{array}$$

Proof

It holds from Lemma 3.2 that$$\begin{array}{c}\hfill \frac{1}{{|x|}^{\mu }}\ast F(u_n)\le \frac{k}{2}.\end{array}$$Then we have that

6.1

$$\begin{array}{c}\hfill \frac{1}{{|x|}^{\mu }}\ast F(v_n)\le \frac{k}{2}.\end{array}$$Observe that $v_n$ is a solution of problem

6.2

$$\begin{array}{cc}& -\mathrm{div}(|\mathrm{\nabla }{\mathrm{v}}_{\mathrm{n}}{|}^{\mathrm{p}-2}\mathrm{\nabla }{\mathrm{v}}_{\mathrm{n}}+\mathcal{U}({\epsilon }_{\mathrm{n}}\mathrm{x}+{\mathrm{y}}_{\mathrm{n}})|\mathrm{\nabla }{\mathrm{v}}_{\mathrm{n}}{|}^{\mathrm{q}-2}\mathrm{\nabla }{\mathrm{v}}_{\mathrm{n}})\hfill \\ \hfill & +V({\epsilon }_{n}x+y_n){(|u|}^{p-2}{v}_n+\mathcal{U}({\epsilon }_{n}x+y_n)|v_n{|}^{q-2}{v}_n)\hfill \\ \hfill & =\left(\frac{1}{{|x|}^{\mu }},,\ast ,G_{(},{\epsilon }_n,x,+,y_n,,,v_n,)\right)g({\epsilon }_{n}x+y_n,v_n)\mathrm{in}{\mathbb{R}}^{\mathrm{N}}.\hfill \end{array}$$Since $\{v_n\}$ is strongly convergent in ${\mathbb{E}}_{{\mathcal{U}}_0}$ and (6.1), we can exploit the Moser iteration argument in [29] to Eq. (6.2) to finish the proof. $\square$

Proof of Theorem 1.1

First, we prove that for any $\epsilon \in (0,{\epsilon }_{\delta })$ and any solution $u_{\epsilon }$ of (3.1) obtained in Lemma 5.4, one has

6.3

$$\begin{array}{c}\hfill \Vert u_{\epsilon }{\Vert }_{L^{\infty }({\mathrm{\Lambda }}_{\epsilon }^c)}<a.\end{array}$$In contrary, assume that there exist ${\epsilon }_n\to 0$ and solution $u_n\in {\widehat{N}}_{{\epsilon }_n}$ of problem (3.1) such that$$\begin{array}{c}\hfill \Vert u_n{\Vert }_{L^{\infty }({\mathrm{\Lambda }}_{{\epsilon }_n}^c)}\ge a.\end{array}$$By virtue of Proposition 5.1, there exists $\{{\widehat{y}}_n\}\subset {\mathbb{R}}^N$ satisfies ${\epsilon }_n{\widehat{y}}_n\to y_0,$ where $y_0\in M.$ Take $v_n(x)=u_n(x+{\widehat{y}}_n).$ Choose $\gamma >0$ such that $B_{\gamma }(y_0)\subset B_{2\gamma }(y_0)\subset \mathrm{\Lambda }.$ Then we have$$\begin{array}{c}\hfill B_{\frac{\gamma }{{\epsilon }_n}}\left(\frac{y_0}{{\epsilon }_n}\right)\subset B_{\frac{2\gamma }{{\epsilon }_n}}\left(\frac{y_0}{{\epsilon }_n}\right)\subset {\mathrm{\Lambda }}_{{\epsilon }_n}.\end{array}$$Thus one can deduce that for $y\in B_{\frac{\gamma }{{\epsilon }_n}}({\widehat{y}}_n),$$$\begin{array}{c}\hfill |y-\frac{y_0}{{\epsilon }_n}|\le |y-{\widehat{y}}_n|+|{\widehat{y}}_n-\frac{y_0}{{\epsilon }_n}|\le \frac{\gamma }{{\epsilon }_n}+\frac{1}{{\epsilon }_n}|{\epsilon }_n{\widehat{y}}_n-y_0|<\frac{2\gamma }{{\epsilon }_n},\end{array}$$which implies that$$\begin{array}{c}\hfill B_{\frac{\gamma }{{\epsilon }_n}}({\widehat{y}}_n)\subset B_{\frac{2\gamma }{{\epsilon }_n}}\left(\frac{y_0}{{\epsilon }_n}\right)\subset {\mathrm{\Lambda }}_{{\epsilon }_n}.\end{array}$$It follows from Proposition 5.1 that there exists $R>0$ such that$$\begin{array}{c}\hfill v_n(x)\le a\mathrm{for}|\mathrm{x}|\ge \mathrm{R}\mathrm{and}\mathrm{n}\in \mathbb{N}.\end{array}$$Since $u_n(x)=v_n(x+{\widehat{y}}_n),$ then for each $n\in \mathbb{N},$$$\begin{array}{c}\hfill u_n(x)\le a\mathrm{for}|\mathrm{x}-{\widehat{\mathrm{y}}}_{\mathrm{n}}|\ge \mathrm{R}.\end{array}$$By the fact $B_R({\widehat{y}}_n)\subset B_{\frac{\gamma }{{\epsilon }_n}}({\widehat{y}}_n)\subset {\mathrm{\Lambda }}_{{\epsilon }_n},$ we can see$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{{\epsilon }_n}^c\subset B_{\frac{\gamma }{{\epsilon }_n}}^c({\widehat{y}}_n)\subset B_R^c({\widehat{y}}_n).\end{array}$$Then there holds that$$\begin{array}{c}\hfill u_n(x)\le a\mathrm{for}\mathrm{any}\mathrm{x}\in {\mathrm{\Lambda }}_{{\epsilon }_{\mathrm{n}}}^{\mathrm{c}}.\end{array}$$We present a contradiction due to (6.3). Then by (6.3) and Lemma 5.4, problem (1.1) admits at least ${\mathrm{cat}}_{{\mathrm{M}}_{\delta }}(\mathrm{M})$ nonnegative solutions.

Last we show the concentration of solutions. Let ${\epsilon }_n\to 0$ and $u_n$ be the solutions of (1.1). By $(g_1),$ there exists $\nu \in (0,a)$ such that$$\begin{array}{c}\hfill g_{{\epsilon }_n}(x,t)\le \frac{V_0}{k}{t}^{p-1}\mathrm{for}\mathrm{x}\in {\mathbb{R}}^{\mathrm{N}}\mathrm{and}\mathrm{t}\in (0,\nu ).\end{array}$$Proceeding as above, there exists $R>0$ such that$$\begin{array}{c}\hfill \Vert u_n{\Vert }_{L^{\infty }(B_R^c({\widehat{y}}_n))}<\nu .\end{array}$$We can infer that$$\begin{array}{c}\hfill \Vert u_n{\Vert }_{L^{\infty }(B_R({\widehat{y}}_n))}\ge \nu .\end{array}$$If this is not true, one has $|u_n{|}_{\infty }\le \nu .$ From Lemma 3.2, we have$$\begin{array}{c}\hfill \frac{1}{{|x|}^{\mu }}\ast G_{{\epsilon }_n}(x,u_n)\le \frac{k}{2}.\end{array}$$Using $⟨{\mathcal{L}}_{{\epsilon }_n}^{\prime }(u_n),u_n⟩=0,$ there holds$$\begin{array}{cc}\hfill \Vert u_n{\Vert }_{p,{\epsilon }_n}^p+{\Vert u_n\Vert }_{q,{\mu }_{{\epsilon }_n}}^q& ={\int }_{{\mathbb{R}}^N}(\frac{1}{{|x|}^{\mu }}\ast G_{{\epsilon }_n}(x,u_n))g_{{\epsilon }_n}(x,u_n)u_n\text{d}x\le \frac{V_0}{2}{|u_n|}_p^p\hfill \\ \hfill & \le \frac{1}{2}{\Vert u_n\Vert }_{p,{\epsilon }_n}^p.\hfill \end{array}$$Then $u_n=0,$ which is a contradiction. Let ${\eta }_{{\epsilon }_n}$ be a global maximum point of $u_n.$ One can derive ${\eta }_{{\epsilon }_n}={\widehat{y}}_n+p_n$ with $|p_n|\le R.$ By the facts ${\epsilon }_n{\widehat{y}}_n\to y_0\in M$ and $|p_n|\le R,$ we have ${\epsilon }_n{\eta }_{{\epsilon }_n}\to y_0.$ since V is continuous, then$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}V({\epsilon }_n{\eta }_{{\epsilon }_n})=V(y_0)=V_0.\end{array}$$$\square$

Author Contributions

J. Zuo and W. Zhang wrote the main manuscript text. All authors reviewed the manuscript.

Funding

Jiabin Zuo was partially supported by the Guangdong Basic and Applied Basic Research Foundation (2024A1515012389).

Data Availability

No datasets were generated or analysed during the current study.

Declarations

Conflict of interest

The authors have no conflict of interest to declare for this article.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

1. Adams, R.A.: Sobolev Spaces. Pure and Applied Mathematics, vol. 65. Academic Press, New York (1975)

2. Alves, CO; Figueiredo, GM. Multiplicity and concentration of positive solutions for a class of quasilinear problems. Adv. Nonlinear Stud.; 2011; 11, 2 pp. 265-294.2810132 [DOI: https://dx.doi.org/10.1515/ans-2011-0203]

3. Alves, CO; Yang, MB. Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method. Proc. R. Soc. Edinb. Sect. A; 2016; 146, pp. 23-58.3457630 [DOI: https://dx.doi.org/10.1017/S0308210515000311]

4. Alves, CO; Gao, FS; Squassina, M; Yang, MB. Singularly perturbed critical Choquard equations. J. Differ. Equ.; 2017; 263, 7 pp. 3943-3988.3670042 [DOI: https://dx.doi.org/10.1016/j.jde.2017.05.009]

5. Ambrosio, V.: Nonlinear Fractional Schrödinger Equations in ${\mathbb{R}}^N$. Frontiers in Elliptic and Parabolic Problems. Birkhäuser/Springer, Cham (2021)

6. Ambrosio, V; Repovš, D. Multiplicity and concentration results for a $(p,q)$-Laplacian problem in ${\mathbb{R}}^N$. Z. Angew. Math. Phys.; 2021; 72, 1 33. [DOI: https://dx.doi.org/10.1007/s00033-020-01466-7]

7. Ambrosio, V. Regularity and Pohozaev identity for the Choquard equation involving the $p$-Laplacian operator. Appl. Math. Lett.; 2023; 145, 4602758 [DOI: https://dx.doi.org/10.1016/j.aml.2023.108742] 108742.

8. Ambrosio, V. The nonlinear $(p,q)$-Schrödinger equation with a general nonlinearity: existence and concentration. J. Math. Pures Appl.; 2023; 178, 9 pp. 141-184.4656291 [DOI: https://dx.doi.org/10.1016/j.matpur.2023.07.008]

9. Arora, R; Fiscella, A; Mukherjee, T; Winkert, P. Existence of ground state solutions for a Choquard double phase problem. Nonlinear Anal. Real World Appl.; 2023; 73, 4583116 [DOI: https://dx.doi.org/10.1016/j.nonrwa.2023.103914] 103914.

10. Costa, GS; Figueiredo, GM. Existence and concentration of positive solutions for a critical $p$&$q$ equation. Adv. Nonlinear Anal.; 2022; 11, 1 pp. 243-267.4286287 [DOI: https://dx.doi.org/10.1515/anona-2020-0190]

11. Damascelli, L. Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Ann. Inst. H. Poincaré. Analyse Non Linéaire; 1998; 15, 4 pp. 493-516.1632933 [DOI: https://dx.doi.org/10.1016/s0294-1449(98)80032-2]

12. del Pino, M; Felmer, PL. Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var.; 1996; 4, 2 pp. 121-137.1379196 [DOI: https://dx.doi.org/10.1007/BF01189950]

13. Gao, F.S., Moroz, V., Yang, M.B., Zhao, S.N.: Construction of infinitely many solutions for a critical Choquard equation via local Pohožaev identities. Calc. Var. 61(6), Paper No. 222 (2022)

14. Ge, B; Yuan, WS. Quasilinear double phase problems with parameter dependent performance on the whole space. Bull. Sci. Math.; 2024; 191, 4700083 [DOI: https://dx.doi.org/10.1016/j.bulsci.2023.103371] 103371.

15. He, CJ; Li, GB. The regularity of weak solutions to nonlinear scalar field elliptic equations containing $p$&$q$-Laplacians. Ann. Acad. Sci. Fenn. Math.; 2008; 33, 2 pp. 337-371.2431370

16. Liu, WL; Dai, GW. Existence and multiplicity results for double phase problem. J. Differ. Equ.; 2018; 265, 9 pp. 4311-4334.3843302 [DOI: https://dx.doi.org/10.1016/j.jde.2018.06.006]

17. Liu, WL; Dai, GW. Multiplicity results for double phase problems in ${\mathbb{R}}^N$. J. Math. Phys.; 2020; 61, 94151422 [DOI: https://dx.doi.org/10.1063/5.0020702] 091508.

18. Liu, WL; Winkert, P. Combined effects of singular and superlinear nonlinearities in singular double phase problems in ${\mathbb{R}}^N$. J. Math. Anal. Appl.; 2022; 507, 2 [DOI: https://dx.doi.org/10.1016/j.jmaa.2021.125762] 125762.

19. Liu, ZH; Papageorgiou, NS. Nonautonomous $(p,q)$-equations with unbalanced growth and competing nonlinearities. J. Math. Pures Appl.; 2024; 182, 9 pp. 164-194.4677156 [DOI: https://dx.doi.org/10.1016/j.matpur.2023.12.008]

20. Lieb, E; Loss, M. Analysis; 2001; Providence, AMS:

21. Marcellini, P. Growth conditions and regularity for weak solutions to nonlinear elliptic pdes. J. Math. Anal. Appl.; 2021; 501, 14258802 [DOI: https://dx.doi.org/10.1016/j.jmaa.2020.124408] 124408.

22. Stegliński, R. Infinitely many solutions for double phase problem with unbounded potential in ${\mathbb{R}}^N$. Nonlinear Anal.; 2022; 214, 4318850 [DOI: https://dx.doi.org/10.1016/j.na.2021.112580] 112580.

23. Szulkin, A; Weth, T. Gao, DY; Montreanu, D. The method of Nehari manifold. Handbook of Nonconvex Analysis and Applications; 2010; Boston, International Press: pp. 597-632.

24. Trudinger, NS. On Harnack type inequalities and their application to quasilinear elliptic equations. Commun. Pure Appl. Math.; 1967; 20, pp. 721-747.226198 [DOI: https://dx.doi.org/10.1002/cpa.3160200406]

25. Willem, M. Minimax Theorems; 1996; Boston, Birkhäuser: [DOI: https://dx.doi.org/10.1007/978-1-4612-4146-1] 24

26. Zeng, SD; Rǎdulescu, VD; Winkert, P. Double phase implicit obstacle problems with convection and multivalued mixed boundary value conditions. SIAM J. Math. Anal.; 2022; 54, 2 pp. 1898-1926.4401799 [DOI: https://dx.doi.org/10.1137/21M1441195]

27. Zhang, J; Zhang, W; Rǎdulescu, VD. Double phase problems with competing potentials: concentration and multiplication of ground states. Math. Z.; 2022; 301, 4 pp. 4037-4078.4449739 [DOI: https://dx.doi.org/10.1007/s00209-022-03052-1]

28. Zhang, J; Zhang, W; Rǎdulescu, VD. Concentrating solutions for singularly perturbed double phase problems with nonlocal reaction. J. Differ. Equ.; 2022; 347, pp. 56-103.4517022 [DOI: https://dx.doi.org/10.1016/j.jde.2022.11.033]

29. Zhang, WQ; Zuo, JB; Rǎdulescu, VD. Concentration of solutions for non-autonomous double-phase problems with lack of compactness. Z. Angew. Math. Phys.; 2024; 75, 4 pp. 1-30.4775797 [DOI: https://dx.doi.org/10.1007/s00033-024-02290-z]

30. Zhang, W.Q., Zuo, J.B., Zhao, P.H.: Multiplicity and concentration of positive solutions for $(p,q)$-Kirchhoff type problems. J. Geom. Anal. 33(5) (2023)

31. Zhang, X; Sun, XQ; Liang, SH; Van, TN. Existence and concentration of solutions to a Choquard equation involving fractional $p$-Laplace via penalization method. J. Geom. Anal.; 2024; 34, 3 90.4694444 [DOI: https://dx.doi.org/10.1007/s12220-023-01516-2]

32. Zhikov, VV. Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat.; 1986; 50, pp. 675-710.864171

33. Zuo, JB; Zhang, WQ; Rǎdulescu, VD. Multiplicity and concentration properties for $(p,q)$-Kirchhoff non-autonomous problems with Choquard nonlinearity. Bull. Sci. Math.; 2024; 191, 4708226 [DOI: https://dx.doi.org/10.1016/j.bulsci.2024.103398] 103398.