Multiple Solutions for Double-Phase Elliptic Problem with NonLocal Interaction

Abstract

This study explores the existence and multiplicity of weak solutions for a double-phase elliptic problem with nonlocal interactions, formulated as a Dirichlet boundary value problem. The associated differential operator exhibits two distinct phases governed by exponents p and q, which satisfy a prescribed structural condition. By employing critical point theory, we establish the existence of at least one weak solution and, under appropriate assumptions, demonstrate the existence of three distinct solutions. The analysis is based on abstract variational methods, with a particular focus on the critical point theorems of Bonanno and Bonanno–Marano.

1. Introduction

Double-phase problems with nonlocal nonlinearities arise in numerous domains of mathematical physics, as they are capable of modeling systems characterized by heterogeneous properties and long-range interactions. These problems effectively describe materials with spatially varying stiffness, such as composites, porous media, or biological tissues—by combining distinct growth conditions within a unified framework. This dual-phase nature proves instrumental in analyzing complex phenomena such as anomalous diffusion, nonlinear elasticity, and phase transitions. Applications extend to electrorheological fluids [1], elasticity theory [2], and Lavrentiev’s phenomenon [3].

The incorporation of nonlocal terms enables the models to capture influences that extend beyond immediate neighborhoods, reflecting real-world behaviors more accurately. Such terms are particularly relevant in fields like electromagnetism, quantum mechanics, and population dynamics, where distant interactions play a significant role. Furthermore, double-phase models have found increasing application in image processing, particularly in image restoration [4,5,6] and super-resolution techniques [7]. Altogether, these models offer a robust approach to studying systems where both local and nonlocal effects significantly impact the dynamics.

The central result of this paper, which establishes the existence of multiple weak solutions, highlights the presence of various equilibrium states or configurations that a physical system can attain under different external influences, such as boundary forces, temperature variations, or external fields. This multiplicity suggests that the system can exhibit multiple stable or metastable states, a key aspect in understanding phase transitions, pattern formation, and bifurcation phenomena across different areas of physics. Consequently, the mathematical analysis of such problems plays a fundamental role in modeling, interpreting, and predicting the behavior of complex physical systems governed by nonuniform and nonlinear interactions.

In [8], Motivated by recent advances in the study of nonlinear double-phase elliptic problems, the authors considered the following parametric Dirichlet problem involving the $(p,q)$-Laplacian:

(1)$\left\{\begin{array}{cc}-\alpha {\Delta }_{p}w-\beta {\Delta }_{q}w=\lambda {\left|w\right|}^{q-2}w,\hfill & \mathrm{in}\Omega ,\hfill \\ w=0,\hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}\right.$

where $\Omega \subset {\mathbb{R}}^N$ is a bounded domain with a $C^2$-boundary, and $\alpha ,\beta ,\lambda >0$ are real parameters. Here, ${\Delta }_{r}w={div\left(\right|Dw|}^{r-2}Dw)$ denotes the r-Laplacian operator for $1<p,q<\infty$ with $p\ne q$. The authors proved that the problem exhibits a continuous spectrum, given by the half-line $(\beta {\widehat{\lambda }}_1\left(q\right),+\infty )$, where ${\widehat{\lambda }}_1\left(q\right)$ represents the principal eigenvalue of the q-Laplacian in $W_0^{1,q}(\Omega )$. As a consequence, for every $\lambda >\beta {\widehat{\lambda }}_1\left(q\right)$, problem $P_{\lambda }$ admits at least one nontrivial solution. Their findings contribute to the spectral analysis of nonhomogeneous differential operators and reveal the impact of mixed growth conditions on the structure of solutions.

Later, in [9], the authors obtained infinitely many distinct positive solutions for the following double-phase problem:

(2)$\left\{\begin{array}{cc}-div\left({|\nabla w|}^{p-2}\nabla w+a\left(x\right){|\nabla w|}^{q-2}\nabla w\right)=g(x,w),\hfill & \mathrm{in}\Omega ,\hfill \\ w=0,\hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}\right.$

where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N$ with $N\ge 2$, and the exponents satisfy $1<p<q<N$. Moreover,

(3)${\displaystyle \frac{q}{p}}<1+{\displaystyle \frac{1}{N}},a:\Omega \to [0,+\infty )\mathrm{is}\mathrm{Lipschitz}\mathrm{continuous},$

and the function $g:\Omega \times \mathbb{R}\to \mathbb{R}$ satisfies the Carathéodory condition. Additionally, there exists $t_0>0$ such that

$\underset{t\in \left[0,t_0\right]}{sup}g(·,t)\in L^{\infty }(\Omega ).$

In [10], the authors explore the existence of weak solutions for a double-phase Dirichlet problem of the following form:

$\left\{\begin{array}{c}-div\left({|\nabla w|}^{p-2}\nabla w+a\left(x\right){|\nabla w|}^{q-2}\nabla w\right)=\lambda \beta \left(x\right)g\left(w\right),\mathrm{in}\Omega ,\hfill \\ w=0,\mathrm{on}\partial \Omega ,\hfill \end{array}\right.$

where $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ with a Lipschitz boundary and dimension $N\ge 2$. The parameters p and q satisfy the conditions

$1<p<N,\mathrm{and}p<q<p^{\ast }:={\displaystyle \frac{Np}{N-p}}.$

Here, $a\in L^{\infty }(\Omega )$ is a non-negative coefficient, $\lambda >0$ represents a real parameter, and $\beta$ belongs to the class

${\Lambda }_{+}(\Omega ):=\left\{\alpha \in L^{\infty }(\Omega )\mid ess\underset{x\in \Omega }{inf}\alpha \left(x\right)>0\right\}.$

The function $g:[0,+\infty )\to \mathbb{R}$ is continuous and satisfies the condition

$\underset{t>0}{sup}G\left(t\right)>0,\mathrm{where}G\left(t\right):={\int }_0^{t}g\left(s\right)ds.$

This study establishes a critical threshold for $\lambda$ in relation to the existence of weak solutions. Specifically, when $0\le \lambda <{\lambda }^{\ast }$, only the trivial solution exists, with ${\lambda }^{\ast }$ explicitly defined in terms of the problem’s parameters. On the other hand, for $\lambda >{\lambda }_{\ast }$, at least two distinct non-negative weak solutions are found, satisfying an energy-related condition. The thresholds ${\lambda }^{\ast }$ and ${\lambda }_{\ast }$ are determined separately, leaving an open problem regarding the existence of solutions within the intermediate range $[{\lambda }^{\ast },{\lambda }_{\ast }]$. These results contribute to a deeper understanding of solution multiplicity in double-phase equations, offering explicit conditions that mark the transition from trivial to multiple solutions.

Furthermore, in [11], the authors studied a boundary value problem involving a double-phase operator with variable exponents and Dirichlet boundary conditions:

$\left\{\begin{array}{c}-div\left({|\nabla w|}^{p\left(x\right)-2}\nabla w+a\left(x\right){|\nabla w|}^{q\left(x\right)-2}\nabla w\right)=\lambda g(x,w)\mathrm{in}\Omega ,\hfill \\ w=0\mathrm{on}\partial \Omega .\hfill \end{array}\right.$

Here, $\Omega \subseteq {\mathbb{R}}^N$ is a bounded domain with a Lipschitz boundary $\partial \Omega$, where $N\ge 2$. The exponents $p,q\in C\left(\overline{\Omega }\right)$ satisfy the conditions

$1<p\left(x\right)<N,p\left(x\right)<q\left(x\right)<{\displaystyle \frac{Np\left(x\right)}{N-p\left(x\right)}}\mathrm{for}\mathrm{all}x\in \overline{\Omega }.$

The function $a\left(x\right)$ is a non-negative element of $L^{\infty }(\Omega )$, and $\lambda >0$ is a real parameter. The function $g:\Omega \times \mathbb{R}\to \mathbb{R}$ satisfies the Carathéodory condition, exhibits subcritical growth, and has a particular asymptotic behavior as $\left|w\right|\to \infty$. Their work establishes the existence of weak solutions under general conditions of subcritical growth and superlinear behavior of the nonlinear term. By employing an abstract critical point theorem, the authors demonstrate the existence of two bounded weak solutions with opposite energy signs. Moreover, in specific cases, these solutions are shown to be non-negative. Further recent developments on double-phase problems can be found in [12,13,14].

Building on previous research, this study examines the multiplicity of weak solutions for a double-phase elliptic problem involving a nonlocal interaction:

$\left\{\begin{array}{c}-div\left({|\nabla w|}^{p-2}\nabla w+\mu \left(x\right){|\nabla w|}^{q-2}\nabla w\right)=\lambda f(x,w){\left({\int }_{\Omega }F(x,w)dx\right)}^{\gamma }\mathrm{in}\Omega ,\hfill \\ w=0\mathrm{on}\partial \Omega .\hfill \end{array}\right.$

Here, $\Omega \subset {\mathbb{R}}^N$ is a bounded domain with a Lipschitz boundary $\partial \Omega$, and $N\ge 2$. The parameter $\gamma$ is a positive constant, and $\mu \in L^{\infty }(\Omega )$ is a non-negative weight. The function $f(x,w)$ satisfies the Carathéodory condition and adheres to the following growth restrictions:

$\left({\mathbf{f}}_{\mathbf{1}}\right)m_1{\left|w\right|}^{\alpha -1}\le f(x,w)\le m_2{\left|w\right|}^{\beta -1},\mathrm{for}\mathrm{a}.\mathrm{e}.(x,w)\in \Omega \times \mathbb{R},$

where $m_1,m_2$ are positive constants, and the exponents satisfy $1\le \alpha \le \beta <p^{\ast }$. The function $F(x,\tau )$ is defined as

$F(x,\tau )={\int }_0^{\tau }f(x,t)dt.$

The exponents p and q satisfy the following conditions:

$\left(\mathbf{H}\right)1\le \beta (\gamma +1)<p<N,\mathrm{and}p<q<p^{\ast }:={\displaystyle \frac{Np}{N-p}}.$

The double-phase operator

$-div\left({|\nabla w|}^{p-2}\nabla w+\mu \left(x\right){|\nabla w|}^{q-2}\nabla w\right),w\in W_0^{1,\mathcal{H}}(\Omega ),$

where $W_0^{1,\mathcal{H}}(\Omega )$ is a Musielak–Orlicz–Sobolev space, is closely associated with a two-phase integral functional.

This paper establishes the existence of a single solution as well as the existence of three distinct solutions, without imposing additional constraints on the exponents p and q beyond the condition (1.2) imposed in [9]. The primary analytical tools employed are critical point theorems as developed in [15,16].

The structure of this paper is as follows: the next section introduces the variational framework and relevant preliminaries, while the final section presents the main results.

2. Variational Framework and Preliminaries

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^N$ with $N\ge 2$ and a Lipschitz boundary $\partial \Omega$. For any $1\le r\le \infty$, we denote by $L^r(\Omega )$ the usual Lebesgue space, equipped with the norm ${\parallel ·\parallel }_r$. When $1\le r<\infty$, we consider the Sobolev spaces $W^{1,r}(\Omega )$ and $W_0^{1,r}(\Omega )$, endowed with the norms ${\parallel ·\parallel }_{1,r}$ and ${\parallel ·\parallel }_{1,r,0}={\parallel \nabla ·\parallel }_r$, respectively.

Define the function $\mathcal{H}:\Omega \times [0,+\infty )\to \mathbb{R}$ by

$\mathcal{H}(x,t)=t^p+\mu \left(x\right)t^q,$

which corresponds to the modular function

${\rho }_{\mathcal{H}}\left(w\right)={\int }_{\Omega }\mathcal{H}(x,|w\left|\right)\mathrm{d}x={\int }_{\Omega }\left({\left|w\right|}^p+\mu \left(x\right){\left|w\right|}^q\right)\mathrm{d}x.$

The associated Musielak–Orlicz space $L^{\mathcal{H}}(\Omega )$ is defined as

$L^{\mathcal{H}}(\Omega )=\left\{w:\Omega \to \mathbb{R}\mathrm{measurable}\mid {\rho }_{\mathcal{H}}\left(w\right)<+\infty \right\},$

which is equipped with the Luxemburg norm

${\parallel w\parallel }_{\mathcal{H}}:=inf\left\{\tau >0\mid {\rho }_{\mathcal{H}}\left({\displaystyle \frac{w}{\tau }}\right)\le 1\right\}.$

The fundamental properties of Musielak–Orlicz spaces play a crucial role in the analysis of functionals with nonstandard growth conditions. For a comprehensive exposition of these properties, we refer the reader to the work of Diening et al. [17].

Proposition 1.

Let p, q satisfy the condition $\left(\mathbf{H}\right)$, let $w\in L^{\mathcal{H}}(\Omega )$, and let $\tau \in \mathbb{R}$. Then, the following properties hold: (1) 

If $w\ne 0$, then

${\parallel w\parallel }_{\mathcal{H}}=\tau ⟺{\rho }_{\mathcal{H}}\left({\displaystyle \frac{w}{\tau }}\right)=1.$

(2) 

${\parallel w\parallel }_{\mathcal{H}}<1$ (respectively $=1$, $>1$) if and only if ${\rho }_{\mathcal{H}}\left(w\right)<1$ (respectively $=1$, $>1$).

(3) 

If ${\parallel w\parallel }_{\mathcal{H}}<1$, then

${\parallel w\parallel }_{\mathcal{H}}^q\le {\rho }_{\mathcal{H}}\left(w\right)\le {\parallel w\parallel }_{\mathcal{H}}^p.$

(4) 

If ${\parallel w\parallel }_{\mathcal{H}}>1$, then

${\parallel w\parallel }_{\mathcal{H}}^p\le {\rho }_{\mathcal{H}}\left(w\right)\le {\parallel w\parallel }_{\mathcal{H}}^q.$

(5) 

${\parallel w\parallel }_{\mathcal{H}}\to 0$ if and only if ${\rho }_{\mathcal{H}}\left(w\right)\to 0$.

(6) 

${\parallel w\parallel }_{\mathcal{H}}\to +\infty$ if and only if ${\rho }_{\mathcal{H}}\left(w\right)\to +\infty$.

(7) 

${\parallel w\parallel }_{\mathcal{H}}\to 1$ if and only if ${\rho }_{\mathcal{H}}\left(w\right)\to 1$.

(8) 

If $w_n\to w$ in $L^{\mathcal{H}}(\Omega )$, then

${\rho }_{\mathcal{H}}\left(w_n\right)\to {\rho }_{\mathcal{H}}\left(w\right).$

The Musielak–Orlicz–Sobolev space $W^{1,\mathcal{H}}(\Omega )$ is defined as

$W^{1,\mathcal{H}}(\Omega ):=\left\{w\in L^{\mathcal{H}}(\Omega ):|\nabla w|\in L^{\mathcal{H}}(\Omega )\right\},$

equipped with the corresponding norm:

${\parallel w\parallel }_{1,\mathcal{H}}:={\parallel \nabla w\parallel }_{\mathcal{H}}+{\parallel w\parallel }_{\mathcal{H}},$

where ${\parallel \nabla w\parallel }_{\mathcal{H}}={\parallel |\nabla w|\parallel }_{\mathcal{H}}$. We denote by $W_0^{1,\mathcal{H}}(\Omega )$ the completion of $C_0^{\infty }(\Omega )$ in $W^{1,\mathcal{H}}(\Omega )$.

It can be shown that both the Musielak–Orlicz and Musielak–Orlicz–Sobolev spaces are uniformly convex (and therefore reflexive) Banach spaces; see Crespo-Blanco et al. [18].

Furthermore, the following embedding results hold.

Proposition 2

(see [18] (Propositions 2.16 and 2.18)). Suppose that $\left(\mathbf{H}\right)$ holds. Then the following are true: (1) 

$W_0^{1,\mathcal{H}}(\Omega )\hookrightarrow L^{\mathcal{H}}(\Omega )$ is compact.

(2) 

$W_0^{1,\mathcal{H}}(\Omega )\hookrightarrow L^s(\Omega )$ is compact for all $s\in \left[1,p^{\ast }\right)$.

(3) 

$W^{1,\mathcal{H}}(\Omega )\hookrightarrow W_0^{1,s}(\Omega )$ is continuous for all $s\in \left[1,p^{\ast }\right]$.

(4) 

$L^{\mathcal{H}}(\Omega )\hookrightarrow L^s(\Omega )$ is continuous for all $s\in \left[1,p^{\ast }\right]$.

Furthermore, a Poincaré-type inequality holds, which allows us to consider in $W_0^{1,\mathcal{H}}(\Omega )$ the equivalent norm

$\parallel w\parallel :={\parallel \nabla w\parallel }_{\mathcal{H}}.$

Let $s<p^{\ast }$ be such that the embedding $W_0^{1,\mathcal{H}}(\Omega )\hookrightarrow L^s(\Omega )$ is continuous. We denote by $c_s$ the best constant for which the following inequality holds:

(4)${\parallel w\parallel }_s\le c_s\parallel w\parallel \forall w\in W_0^{1,\mathcal{H}}(\Omega )$

In other words, $c_s$ is the operator norm of the embedding $i:\left(W_0^{1,\mathcal{H}}(\Omega ),\parallel ·\parallel \right)\to \left(L^s(\Omega ),{\parallel ·\parallel }_s\right)$.

The differential operator in $\left(P_{\lambda }\right)$ is the so-called double-phase operator:

$-div\left({|\nabla w|}^{p-2}\nabla w+\mu \left(x\right){|\nabla w|}^{q-2}\nabla w\right),w\in W_0^{1,\mathcal{H}}(\Omega ).$

Definition 1. 

Let $\lambda \in \mathbb{R}$ be fixed, and by a weak solution to $\left(P_{\lambda }\right)$, we mean a function $w\in W_0^{1,\mathcal{H}}(\Omega )$, such that

(5) ${\int }_{\Omega }\left({|\nabla w|}^{p-2}\nabla w+\mu \left(x\right){|\nabla w|}^{q-2}\nabla w\right)·\nabla \phi \mathrm{d}x=\lambda {\int }_{\Omega }f(x,w)\phi dx{\left({\int }_{\Omega }F(x,w)dx\right)}^{\gamma }$

Let us define the functional ${\mathcal{I}}_{\lambda }:W_0^{1,\mathcal{H}}(\Omega )\to \mathbb{R}$ as

${\mathcal{I}}_{\lambda }\left(w\right):=\mathcal{L}\left(w\right)-\lambda \mathcal{M}\left(w\right),$

where

(6)$\begin{array}{c}\hfill \mathcal{L}\left(w\right):={\int }_{\Omega }\left({\displaystyle \frac{{|\nabla w|}^p}{p}}+\mu \left(x\right){\displaystyle \frac{{|\nabla w|}^q}{q}}\right)\mathrm{d}x,\end{array}$

and

(7)$\begin{array}{c}\hfill \mathcal{M}\left(w\right):={\displaystyle \frac{1}{\gamma +1}}{\left({\int }_{\Omega }F(x,w)dx\right)}^{\gamma +1}.\end{array}$

It is easy to see that, under condition $\left(\mathbf{H}\right)$ and $\left({\mathbf{f}}_{\mathbf{1}}\right)$, $\mathcal{L}$ and $\mathcal{M}$ are well defined and continuously Gâteaux differentiable with

$⟨{\mathcal{L}}^{\prime }\left(w\right),\phi ⟩={\int }_{\Omega }\left({|\nabla w|}^{p-2}\nabla w·\nabla \phi +\mu \left(x\right){|\nabla w|}^{q-2}\nabla w·\nabla \phi \right)dx,$

and

${\displaystyle ⟨{\mathcal{M}}^{\prime }\left(w\right),\phi ⟩={\left({\int }_{\Omega }F(x,w)dx\right)}^{\gamma }{\int }_{\Omega }f(x,w)\phi dx.}$

Lemma 1 

(see [19], (Proposition 3.1)).

The functional ${\mathcal{L}}^{\prime }$ is strictly monotone in $W_0^{1,\mathcal{H}}(\Omega ).$

The functional ${\mathcal{L}}^{\prime }$ is a mapping of $\left(S_{+}\right)$ type, i.e., if $w_n\rightharpoonup w$ in $W_0^{1,\mathcal{H}}(\Omega ),$ and ${{\displaystyle \overline{lim}}}_{n\to \infty }⟨{\mathcal{L}}^{\prime }\left(w_n\right)-{\mathcal{L}}^{\prime }\left(w\right),w_n-w)⟩\le 0,$ then $w_n\to w$ in $W_0^{1,\mathcal{H}}(\Omega ).$

The functional ${\mathcal{L}}^{\prime }$ is a homeomorphism.

Lemma 2. 

The functional ${\mathcal{M}}^{\prime }:X:=W_0^{1,\mathcal{H}}(\Omega )\to {\left(W_0^{1,\mathcal{H}}(\Omega )\right)}^{\ast }$ is compact.

Proof. 

Condition $\left({\mathbf{f}}_{\mathbf{1}}\right)$, together with the compact embeddings $X\hookrightarrow L^{\beta }(\Omega )$, where $1\le \beta <p^{\ast }$ (see $\left(2\right)$, in Proposition 2), imply that ${\mathcal{M}}^{\prime }\left(w\right)$ is compact. In fact, let us consider a sequence $\left\{w_k\right\}\subset X$ such that $w_k\rightharpoonup u$. The embedding $X\hookrightarrow L^{\beta }(\Omega )$, where $1\le \beta <p^{\ast }$, is compact; therefore, there exists a subsequence of $\left\{w_k\right\}$, which we still denote by $\left\{w_k\right\}$, such that $w_k\to w$ strongly in $L^{\beta }(\Omega )$ and $w_k\left(x\right)\to w\left(x\right)$ for almost every $x\in \Omega$. The continuity of $F(x,w)$ with respect to w ensures that

$F(x,w_k)\to F(x,w)\mathrm{for} \mathrm{almost} \mathrm{every}x.$

Now, since $F(x,\tau )={\int }_0^{\tau }f(x,t)dt$, then, from condition $\left({\mathbf{f}}_{\mathbf{1}}\right)$, one has

$|F(x,w_k)|<{\displaystyle \frac{m_2}{\beta }}{\left|w_k\right|}^{\beta }.$

By using the dominated Convergence theorem, we can write

(8)$\begin{array}{c}\hfill {\displaystyle {\int }_{\Omega }F(x,w_k)dx\to {\int }_{\Omega }F(x,w_k)dxask\to +\infty .}\end{array}$

Then, from the condition $\left({\mathbf{f}}_{\mathbf{1}}\right)$, we affirm the continuity of the Nemytskii operator $N_f\left(w\right)\left(x\right)=f(x,w\left(x\right))$, as $f:\Omega \times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function that satisfies condition $\left({\mathbf{f}}_{\mathbf{1}}\right)$, and then $N_f\left(w_k\right)\to N_f\left(w\right)$ in $L^{{\displaystyle \frac{\beta }{\beta -1}}}(\Omega )$. Next, by utilizing the Hölder inequality, for any $\phi \in E$, we have the estimate

(9)$\begin{array}{ccc}\hfill \left|{\int }_{\Omega }f(x,w_k)\phi dx-{\int }_{\Omega }f(x,w)\phi dx\right|& \le & {\int }_{\Omega }\left|(f(x,w_k)-f(x,w))\phi \right|dx\hfill \\ & \le & 2\parallel N_f\left(w_k\right)-N_f\left(w\right){\parallel }_{\frac{\beta }{\beta -1}}{\parallel \phi \parallel }_{\beta },\hfill \\ & \le & 2c_{\beta }\parallel N_f\left(w_k\right)-N_f\left(w\right){\parallel }_{\frac{\beta }{\beta -1}}\parallel \phi \parallel \hfill \end{array}$

where $c_{\beta }$ is the constant associated with the embedding $X\hookrightarrow L^{\beta }(\Omega )$ with $1\le \beta <p^{\ast }$. By combining the results from Equations (8) and (9), we deduce that ${\mathcal{M}}^{\prime }\left(w_k\right)\to {\mathcal{M}}^{\prime }\left(w\right)$ in $X^{\ast }$, which implies that ${\mathcal{M}}^{\prime }$ is completely continuous. Therefore, ${\mathcal{M}}^{\prime }$ is compact. □

The forthcoming definition and critical point theorems serve as fundamental tools in establishing our main results. To formulate our existence theorem, we first introduce the necessary preliminary definitions and key theorems.

Definition 2. 

Let $\mathcal{L}$ and $\mathcal{M}$ be two continuously Gâteaux differentiable functionals defined on a real Banach space X and fix $d\in \mathbb{R}$. The functional $\mathcal{I}:=\mathcal{L}-\mathcal{M}$ is said to verify the Palais–Smale condition cut of upper at d (in short, ${\left(PS\right)}^{\left[d\right]}$) if any sequence ${\left\{w_n\right\}}_{n\in \mathbb{N}}\in X$ such that the following apply:

$\mathcal{I}\left(w_n\right)$ is bounded;

${lim}_{n\to +\infty }{∥{\mathcal{I}}^{\prime }\left(w_n\right)∥}_{X^{\ast }}=0$;

$\mathcal{L}\left(w_n\right)<d$ for each $n\in \mathbb{N}$ has a convergent subsequence.

If $d=\infty$, the functional $\mathcal{I}:=\mathcal{L}-\mathcal{M}$ satisfies the Palais–Smale condition.

Our main existence result is due to the following Theorem.

Theorem 1 

(Theorem 3.2 [15]). Let X be a real Banach space, and let $\mathcal{L},\mathcal{M}:X\to \mathbb{R}$ be two continuously Gâteaux differentiable functionals such that

$\left(i\right)\underset{x\in X}{inf}\mathcal{L}=\mathcal{L}\left(0\right)=\mathcal{M}\left(0\right)=0.$

Assume that there exists the positive constant $d\in \mathbb{R}$ and $\overline{x}\in X$ with $0<\mathcal{L}\left(\overline{x}\right)<d$ such that

$\left(ii\right){\displaystyle \frac{{\displaystyle \underset{x\in {\mathcal{L}}^{-1}\left(\right]-\infty ,d\left[\right)}{sup}\mathcal{M}\left(x\right)}}{d}}<{\displaystyle \frac{\mathcal{M}\left(\overline{x}\right)}{\mathcal{L}\left(\overline{x}\right)}}$

and

$\left(iii\right)for any\lambda \in \Lambda :=]{\displaystyle \frac{\mathcal{L}\left(\overline{x}\right)}{\mathcal{M}\left(\overline{x}\right)}},{\displaystyle \frac{d}{{\displaystyle \underset{x\in {\mathcal{L}}^{-1}\left(\right]-\infty ,d\left[\right)}{sup}\mathcal{M}\left(x\right)}}}[,{\mathcal{I}}_{\lambda }=\mathcal{L}-\lambda \mathcal{M}satisfies{\left(PS\right)}^{\left[d\right]}-condition.$

Then, for every $\lambda \in \Lambda$, there is $w_{\lambda }\in$${\mathcal{L}}^{-1}\left(\right]0,d\left[\right)$ such that ${\mathcal{I}}_{\lambda }\left(w_{\lambda }\right)\le {\mathcal{I}}_{\lambda }\left(w\right)$ for all $w\in {\mathcal{L}}^{-1}\left(\right]0,d\left[\right)$ and ${\mathcal{I}}_{\lambda }^{\prime }\left(w_{\lambda }\right)=0$.

Theorem 2 

([16] Theorem 3.6). Let X be a reflexive real Banach space and assume the following:

$\mathcal{L}:X\to \mathbb{R}$ is a coercive functional that is continuously Gateaux differentiable and weakly lower semicontinuous in the sequential sense.

The Gateaux derivative of $\mathcal{L}$ has a continuous inverse on the dual space $X^{\ast }$.

$\mathcal{M}:X\to \mathbb{R}$ is a continuously Gateaux differentiable functional with a compact Gateaux derivative.

Furthermore, suppose that

$\left(a_{\mathit{0}}\right)\underset{X}{inf}\mathcal{L}=\mathcal{L}\left(0\right)=0and\mathcal{M}\left(0\right)=0.$

There exists a positive constant d and a point $\overline{v}\in X$ such that $d<\mathcal{L}\left(\overline{v}\right)$, and the following conditions are satisfied:

$\left(a_{\mathit{1}}\right){\displaystyle \frac{{sup}_{\mathcal{L}\left(x\right)<d}\mathcal{M}\left(x\right)}{d}}<{\displaystyle \frac{\mathcal{M}\left(\overline{v}\right)}{\mathcal{L}\left(\overline{v}\right)}},$

$\left(a_{\mathit{2}}\right)\mathit{For}\mathit{each}\lambda \in {\Lambda }_d:=\left({\displaystyle \frac{\mathcal{L}\left(\overline{v}\right)}{\mathcal{M}\left(\overline{v}\right)}},{\displaystyle \frac{d}{{sup}_{\mathcal{L}\left(x\right)\le d}\mathcal{M}\left(x\right)}}\right),\mathit{the}\mathit{functional}{I}_{\lambda }:=\mathcal{L}-\lambda \mathcal{M}\mathit{is}\mathit{coercive}.$

Then, for any $\lambda \in {\Lambda }_d$, $\mathcal{L}-\lambda \mathcal{M}$ has at least three distinct critical points in X.

3. Main Results

In this section, a theorem about the existence of at least three weak solutions to the problem $\left(P_{\lambda }\right)$ is obtained. First, we mention that, for a large enough $\parallel w\parallel$, and due to condition $\left(\mathbf{H}\right)$ and $\left(4\right)$ in Proposition 1, one has

(10)$\begin{array}{cc}\hfill \mathcal{L}\left(w\right):& ={\int }_{\Omega }\left({\displaystyle \frac{{|\nabla w|}^p}{p}}+\mu \left(x\right){\displaystyle \frac{{|\nabla w|}^q}{q}}\right)\mathrm{d}x\hfill \\ \hfill & \ge {\displaystyle \frac{1}{q}}{\rho }_{\mathcal{H}}(\nabla w)\hfill \\ \hfill & \ge {\displaystyle \frac{1}{q}}{\parallel w\parallel }^p;\hfill \end{array}$

thus, $\mathcal{L}$ is bounded from below. Furthermore, one has the following.

Lemma 3. 

Assume that conditions $\left({\mathbf{f}}_{\mathbf{1}}\right)$ and $\left(\mathbf{H}\right)$ hold, then ${\mathcal{I}}_{\lambda }$ satisfies the Palais–Smale condition for any $\lambda >0$.

Proof. 

Let $\left\{w_n\right\}\subseteq X$ be a Palais–Smale sequence, so, one has

(11)$\underset{n}{sup}{\mathcal{I}}_{\lambda }\left(w_n\right)<+\infty \mathrm{and}{∥{\mathcal{I}}_{\lambda }^{\prime }\left(w_n\right)∥}_{X^{\ast }}⟶0$

Let us show that $\left\{w_n\right\}\subseteq X$ contains a convergent subsequence. By condition $\left({\mathbf{f}}_{\mathbf{1}}\right)$, one has

$\begin{array}{cc}\hfill ⟨{\mathcal{M}}^{\prime }\left(w\right),w⟩& {\displaystyle ={\left({\int }_{\Omega }F(x,w)dx\right)}^{\gamma }{\int }_{\Omega }f(x,w)wdx,}\hfill \\ & \le {\left({\int }_{\Omega }{\displaystyle \frac{m_2}{\beta }}{\left|w\right|}^{\beta }dx\right)}^{\gamma }{\int }_{\Omega }{m}_2{\left|w\right|}^{\beta }dx,\hfill \\ & \le {\displaystyle \frac{m_2^{\gamma +1}}{{\beta }^{\gamma }}}{\parallel w\parallel }_{\beta }^{\beta (\gamma +1)},\hfill \\ & \le {\displaystyle \frac{m_2^{\gamma +1}{c}_{\beta }^{\beta (\gamma +1)}}{{\beta }^{\gamma }}}{\parallel w\parallel }^{\beta (\gamma +1)}.\hfill \end{array}$

where $c_{\beta }$ is the constant from the continuous embedding of X into $L^{\beta }(\Omega )$.

Then, for n that is large enough, one has

$\begin{array}{cc}\hfill ⟨{\mathcal{I}}_{\lambda }^{\prime }\left(w_n\right),w_n⟩& =⟨{\mathcal{L}}_{\lambda }^{\prime }\left(w_n\right),w_n⟩-\lambda ⟨{\mathcal{M}}_{\lambda }^{\prime }\left(w_n\right),w_n⟩\hfill \\ & \ge \parallel w_n{\parallel }^p-\lambda {\displaystyle \frac{m_2^{\gamma +1}{c}_{\beta }^{\beta (\gamma +1)}}{{\beta }^{\gamma }}}{\parallel w_n\parallel }^{\beta (\gamma +1)}.\hfill \end{array}$

Moreover, using (11), we have

$\parallel w_n{\parallel }^p\le \lambda {\displaystyle \frac{m_2^{\gamma +1}{c}_{\beta }^{\beta (\gamma +1)}}{{\beta }^{\gamma }}}{\parallel w_n\parallel }^{\beta (\gamma +1)},$

since $\beta (\gamma +1)<p$, then $\left\{w_n\right\}$ is bounded. Passing to a subsequence if necessary, we can assume that $w_n\rightharpoonup w$; thus, ${\mathcal{M}}^{\prime }\left(w_n\right)\to {\mathcal{M}}^{\prime }\left(w\right)$ because of the compactness of ${\mathcal{M}}^{\prime }.$ Combining with ${\mathcal{I}}_{\lambda }^{\prime }\left(w_n\right)={\mathcal{L}}^{\prime }\left(w_n\right)-\lambda {\mathcal{M}}^{\prime }\left(w_n\right)\to 0,$ one has ${\mathcal{L}}^{\prime }\left(w_n\right)\to \lambda {\mathcal{M}}^{\prime }\left(w\right).$ Since ${\mathcal{L}}^{\prime }$ is a homeomorphism, then $w_n\to w.$ Thus, ${\mathcal{I}}_{\lambda }$ satisfies the Palais–Smale condition. □

We are now ready to present our main result. To this end, we define

$\tilde{\mathfrak{D}}\left(x\right):=sup\left\{\tilde{\mathfrak{D}}>0\mid B(x,\tilde{\mathfrak{D}})\subseteq \Omega \right\}$

for each $x\in \Omega$, where $B(x,\tilde{\mathfrak{D}})$ denotes a ball centered at x with radius $\tilde{\mathfrak{D}}$. It is evident that there exists a point $x^0\in \Omega$ such that $B(x^0,R)\subseteq \Omega$, where

$R=\underset{x\in \Omega }{sup}\tilde{\mathfrak{D}}\left(x\right).$

In what follows, the symbol $\tilde{m}$ will represent the constant

$\tilde{m}={\displaystyle \frac{{\pi }^{\frac{N}{2}}}{\frac{N}{2}\Gamma \left(\frac{N}{2}\right)}},$

with $\Gamma$ denoting the Gamma function.

Theorem 3. 

Assume conditions $\left({\mathbf{f}}_{\mathbf{1}}\right)$ and $\left(\mathbf{H}\right)$ hold; moreover, suppose that there exist two positive constants d and $\delta >0$, such that

(12) $\begin{array}{c}\hfill {\displaystyle \frac{1}{p}}\left({\left({\displaystyle \frac{2\delta }{R}}\right)}^p+{\left({\displaystyle \frac{2\delta }{R}}\right)}^q{\parallel \mu \parallel }_{\infty }\right)\tilde{m}\left(R^N-{\left({\displaystyle \frac{R}{2}}\right)}^N\right)<d,\end{array}$

and

$A_{\delta }:={\displaystyle \frac{\frac{1}{p}\left({\left(\frac{2\delta }{R}\right)}^p+{\left(\frac{2\delta }{R}\right)}^q{\parallel \mu \parallel }_{\infty }\right)\tilde{m}\left(R^N-{\left(\frac{R}{2}\right)}^N\right)}{\frac{m_1^{\gamma +1}}{(\gamma +1){\alpha }^{\gamma +1}}{\left({\delta }^{\alpha }\tilde{m}{\left(\frac{R}{2}\right)}^N\right)}^{\gamma +1}}}<B_d:={\displaystyle \frac{d}{\frac{c_{\beta }^{\beta (\gamma +1)}{m}_2^{\gamma +1}}{(\gamma +1){\beta }^{\gamma +1}}max\left\{{\left(dq\right)}^{\frac{\beta (\gamma +1)}{p}},{\left(dq\right)}^{\frac{\beta (\gamma +1)}{q}}\right\}}};$

then, for any $\lambda \in ]A_{\delta },B_d[$, problem $\left(P_{\lambda }\right)$ has at least one weak solution $w_{\lambda }$, such that, $\mathcal{L}\left(w_{\lambda }\right)\in ]0,d[$, and ${\mathcal{I}}_{\lambda }^{\prime }\left(w_{\lambda }\right)=0$$\left(w_{\lambda }nontrivial\right)$.

Proof. 

As we showed previously, the functional $\mathcal{M}$ and $\mathcal{L}$ are continuously Gâteaux differentiable; moreover, condition $\left(i\right)$ of Theorem 1 holds. Let d and $\delta$ be as in (12), and define $v_d\in X$ such that

$v_{\delta }\left(x\right):=\left\{\begin{array}{cc}0\hfill & x\in \Omega \setminus B\left(x^0,R\right)\hfill \\ {\displaystyle \frac{2\delta }{R}}\left(R-\left|x-x^0\right|\right)\hfill & x\in B\left(x^0,R\right)\setminus B\left(x^0,{\displaystyle \frac{R}{2}}\right),\hfill \\ \delta \hfill & x\in B\left(x^0,{\displaystyle \frac{R}{2}}\right).\hfill \end{array}\right.$

Then, by the definition of the functional $\mathcal{L}$, we have

(13)$\begin{array}{c}\hfill \begin{array}{cc}& {\displaystyle \frac{1}{p}}{\left({\displaystyle \frac{2\delta }{R}}\right)}^p\tilde{m}\left(R^N-{\left({\displaystyle \frac{R}{2}}\right)}^N\right)\hfill \\ & <\mathcal{L}\left(v_{\delta }\right)\hfill \\ & \le {\displaystyle \frac{1}{p}}\left({\left({\displaystyle \frac{2\delta }{R}}\right)}^p+{\left({\displaystyle \frac{2\delta }{R}}\right)}^q{\parallel \mu \parallel }_{\infty }\right)\tilde{m}\left(R^N-{\left({\displaystyle \frac{R}{2}}\right)}^N\right).\hfill \end{array}\end{array}$

Therefore, $0<\mathcal{L}\left(v_{\delta }\right)<d$, and together with Lemma 3, we can deduce that ${\mathcal{I}}_{\lambda }$ satisfies ${\left(PS\right)}^{\left[d\right]}$-condition. Moreover, by the definition of $\mathcal{M}$, the expression of $v_{\delta }$ and the assumption $\left({\mathbf{f}}_{\mathbf{1}}\right)$, one has

(14)$\begin{array}{c}\hfill \mathcal{M}\left(v_{\delta }\right)\ge {\displaystyle \frac{1}{\gamma +1}}{\left({\int }_{B\left(x_0,{\displaystyle \frac{R}{2}}\right)}{\displaystyle \frac{m_1}{\alpha }}{\left|v_{\delta }\right|}^{\alpha }dx\right)}^{\gamma +1}\ge {\displaystyle \frac{m_1^{\gamma +1}}{(\gamma +1){\alpha }^{\gamma +1}}}{\left({\delta }^{\alpha }\tilde{m}{\left({\displaystyle \frac{R}{2}}\right)}^N\right)}^{\gamma +1}.\end{array}$

This yields to

${\displaystyle \frac{\mathcal{M}\left(v_{\delta }\right)}{\mathcal{L}\left(v_{\delta }\right)}}\ge {\displaystyle \frac{\frac{m_1^{\gamma +1}}{(\gamma +1){\alpha }^{\gamma +1}}{\left({\delta }^{\alpha }\tilde{m}{\left(\frac{R}{2}\right)}^N\right)}^{\gamma +1}}{\frac{1}{p}\left({\left(\frac{2\delta }{R}\right)}^p+{\left(\frac{2\delta }{R}\right)}^q{\parallel \mu \parallel }_{\infty }\right)\tilde{m}\left(R^N-{\left(\frac{R}{2}\right)}^N\right)}}.$

In addition, for each $w\in {\mathcal{L}}^{-1}\left(\right]-\infty ,d\left[\right)$, we have

(15)${\displaystyle \frac{1}{q}}min\left({\parallel w\parallel }^p,{\parallel w\parallel }^q\right)\le d.$

Therefore,

$\parallel w\parallel \le max\left\{{\left(dq\right)}^{{\displaystyle \frac{1}{p}}},{\left(dq\right)}^{{\displaystyle \frac{1}{q}}}\right\},$

Finally, we have following result:

$\begin{array}{ccc}\hfill {\displaystyle \underset{\mathcal{L}\left(w\right)<d}{sup}\mathcal{M}\left(w\right)}& \le & \underset{\mathcal{L}\left(w\right)<d}{sup}{\displaystyle \frac{m_2^{\gamma +1}}{(\gamma +1){\beta }^{\gamma +1}}}{\parallel w\parallel }_{\beta }^{\beta (\gamma +1)},\hfill \\ & \le & {\displaystyle \frac{c_{\beta }^{\beta (\gamma +1)}{m}_2^{\gamma +1}}{(\gamma +1){\beta }^{\gamma +1}}}max\left\{{\left(dq\right)}^{\frac{\beta (\gamma +1)}{p}},{\left(dq\right)}^{\frac{\beta (\gamma +1)}{q}}\right\}\hfill \end{array}$

and

$\begin{array}{c}\hfill {\displaystyle \frac{1}{d}}\underset{\mathcal{L}\left(w\right)<d}{sup}\mathcal{M}\left(w\right)<{\displaystyle \frac{1}{\lambda }}.\end{array}$

This completes the proof. □

Theorem 4. 

Assume conditions $\left({\mathbf{f}}_{\mathbf{1}}\right)$ and $\left(\mathbf{H}\right)$ hold; moreover, suppose that there exist two positive constants d and $\delta >0$, such that

${\displaystyle \frac{1}{p}}{\left({\displaystyle \frac{2\delta }{R}}\right)}^p\tilde{m}\left(R^N-{\left({\displaystyle \frac{R}{2}}\right)}^N\right)=d;$

then, for any $\lambda \in ]A_{\delta },B_d[$, ($A_{\delta }and{B}_d$ are those of Theorem 3), problem $\left(P_{\lambda }\right)$ has at least three weak solutions.

Proof. 

It is important to note that the functionals $\mathcal{L}$ and $\mathcal{M}$, associated with problem $\left(P_{\lambda }\right)$ and defined in (6) and (7), satisfy the regularity conditions outlined in Theorem 2. We now proceed to establish the fulfillment of conditions $\left(a_1\right)$ and $\left(a_2\right)$. To this end, consider

${\displaystyle \frac{1}{p}}{\left({\displaystyle \frac{2\delta }{R}}\right)}^p\tilde{m}\left(R^N-{\left({\displaystyle \frac{R}{2}}\right)}^N\right)=d.$

From inequality (13), it follows that $d<\mathcal{L}\left(v_{\delta }\right)$. Additionally, we can establish the coerciveness of ${\mathcal{I}}_{\lambda }$ for any positive value of $\lambda$ by employing inequality (14). For sufficiently large $\parallel w\parallel$, we deduce that

$\mathcal{L}\left(w\right)-\lambda \mathcal{M}\left(w\right)\ge {\displaystyle \frac{1}{q}}{\parallel w\parallel }^p-\lambda {\displaystyle \frac{m_2^{\gamma +1}}{(\gamma +1){\beta }^{\gamma +1}}}{\parallel w\parallel }_{\beta }^{\beta (\gamma +1)}.$

Since $p>\beta (\gamma +1)$, we can reach the desired conclusion. Finally, considering the fact that

${\overline{\Lambda }}_d:=\left(A_{\delta },B_d\right)\subseteq \left({\displaystyle \frac{\mathcal{L}\left(v_{\delta }\right)}{\mathcal{M}\left(v_{\delta }\right)}},{\displaystyle \frac{d}{{sup}_{\mathcal{L}\left(w\right)<d}\mathcal{M}\left(u\right)}}\right),$

and noting that all assumptions of Theorem 2 are satisfied, it follows that for any $\lambda \in {\overline{\Lambda }}_d$, the functional $\mathcal{L}-\lambda \mathcal{M}$ possesses at least three critical points in $X:=W_0^{\mathcal{H}}(\Omega )$. Consequently, these critical points are precisely the weak solutions of problem $\left(P_{\lambda }\right)$. □

4. Conclusions

The study presented in this document explores the existence and multiplicity of weak solutions for a double-phase elliptic problem with nonlocal interactions, demonstrating that under specific conditions, at least one weak solution exists, and up to three distinct solutions may arise. This multiplicity highlights the complex behavior of physical systems influenced by nonlocal interactions, which can lead to various stable configurations. A promising future direction for this research is to extend the analysis to nonstandard growth conditions. This modification allows for a more nuanced understanding of materials with spatially varying properties. The problem can then be formulated as

$\left\{\begin{array}{cc}-\mathrm{div}\left({|\nabla w|}^{p\left(x\right)}\nabla w+a\left(x\right){|\nabla w|}^{q\left(x\right)}\nabla w\right)=\lambda f(x,w){\left({\int }_{\Omega }F(x,w)dx\right)}^{\gamma },\hfill & \mathrm{in}\Omega ,\hfill \\ w=0,\hfill & \mathrm{on}\partial \Omega .\hfill \end{array}\right.$

This formulation will enable a comprehensive exploration of the implications of variable growth on the solution’s existence and multiplicity, further enriching the understanding of phase transitions and pattern formation in materials characterized by heterogeneous properties.

Author Contributions

Investigation and formal analysis, K.K. and M.M.A.-S.; writing—original draft, writing—review and editing, K.K. and M.M.A.-S. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding authors.

Acknowledgments

The authors extend their appreciation to Northern Border University, Saudi Arabia, for supporting this work through project number (NBU-CRP-2025-1706).

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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