Existence and Related Properties of Solutions for Klein–Gordon–Maxwell Systems

Abstract

This paper aims to present the application of variational methods in studying the existence and related properties of solutions for Klein–Gordon–Maxwell systems through a literature review. First, we give an introduction and variational framework for the Klein–Gordon–Maxwell system. Second, we review the existence and nonexistence of solutions for autonomous Klein–Gordon–Maxwell systems under subcritical growth, critical growth and zero-mass conditions. Third, we introduce studies on the existence and properties of solutions for Klein–Gordon–Maxwell systems classified according to potential functions. Finally, we review the existence of solutions for Klein–Gordon–Maxwell systems in two-dimensional space.

1. Introduction

The existence of solitary waves for scalar fields in dimension 3 has been extensively studied by many authors. The equation they have considered is a nonlinear perturbation of the Klein–Gordon equation. A typical simple case is the following one:

(1)$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^2\psi }{\partial t^2}}-\Delta \psi +m^2\psi -{\left|\psi \right|}^{p-2}\psi =0,\end{array}$

where $\psi =\psi (x,t)\in \mathcal{C}({\mathbb{R}}^3,\mathbb{R})$. The Lagrangian density relative to (1) is given by

(2)$\begin{array}{c}\hfill {\mathcal{L}}_{\mathcal{KG}}={\displaystyle \frac{1}{2}}\left[{\left|{\displaystyle \frac{\partial \psi }{\partial t}}\right|}^2-{|\nabla \psi |}^2-m^2{\left|\psi \right|}^2\right]+{\displaystyle \frac{1}{p}}{\left|\psi \right|}^p.\end{array}$

In recent years, many papers have been devoted to finding standing waves of (1), i.e., solutions of the form

$\begin{array}{c}\hfill \psi (x,t)={\mathrm{e}}^{i\omega t}u\left(x\right),\omega \in \mathbb{R}.\end{array}$

Standing waves have drawn significant attention in both mathematical and physical research. For example, standing waves are utilized in musical instrument manufacturing in practical applications. Under the form of standing waves, the nonlinear Klein–Gordon equation is reduced to a semilinear elliptic equation; then we can use the variational method to find its standing wave solutions.

Next, we explore the interaction of nonlinear Klein–Gordon fields interacting with the electromagnetic field $(\mathbf{E},\mathbf{H})$. Since $(\mathbf{E},\mathbf{H})$ are not assigned, we have to study a system of equations whose unknowns are the Klein–Gordon field $\psi (x,t)$ and the gauge potentials $\mathbf{A}=\mathbf{A}(x,t), \varphi =\varphi (x,t)$ related to the electromagnetic field, where

$\begin{array}{c}\hfill \mathbf{A}:{\mathbb{R}}^3\times \mathbb{R}\to \mathbb{R},\varphi :{\mathbb{R}}^3\times \mathbb{R}\to \mathbb{R}\end{array}$

are related to the electromagnetic field $(\mathbf{E},\mathbf{H})$ by the Maxwell equations

$\begin{array}{c}\hfill \mathbf{E}=-\left(▽\varphi +{\displaystyle \frac{\partial \mathbf{A}}{\partial t}}\right),\mathbf{H}=\nabla \times \mathbf{A}.\end{array}$

By gauge invariance arguments, the interaction between $\psi$ and the electromagnetic field $(\mathbf{E},\mathbf{H})$ is usually described substituting in (2) the usual derivatives ${\displaystyle \frac{\partial }{\partial t}},\nabla$ with the gauge covariant derivatives

$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}\mapsto {\displaystyle \frac{\partial }{\partial t}}+i\mathrm{e}\varphi ,\nabla \mapsto \nabla -i\mathrm{e}\mathbf{A},\end{array}$

where $\mathrm{e}$ denotes the electric charge. Then we have the following Lagrangian density:

(3)$\begin{array}{c}\hfill {\mathcal{L}}_{\mathcal{KGM}}={\displaystyle \frac{1}{2}}\left[{\left|{\displaystyle \frac{\partial \psi }{\partial t}}+i\mathrm{e}\varphi \psi \right|}^2{-|\nabla \psi -i\mathrm{e}\mathbf{A}\psi |}^2-m^2{\left|\psi \right|}^2\right]+{\displaystyle \frac{1}{p}}{\left|\psi \right|}^p.\end{array}$

Set

$\psi (x,t)=u(x,t){\mathrm{e}}^{iS(x,t)},u,S:{\mathbb{R}}^3\times \mathbb{R}\to \mathbb{R},$

The Lagrangian density (3) takes the following form:

$\begin{array}{c}\hfill {\mathcal{L}}_{\mathcal{KGM}}={\displaystyle \frac{1}{2}}\left\{u_t^2-{|\nabla u|}^2{-\left[\right|\nabla S-\mathrm{e}\mathbf{A}|}^2-{(S_t+\mathrm{e}\varphi )}^2+m^2{\left]\right|u|}^2\right\}+{\displaystyle \frac{1}{p}}{\left|u\right|}^p.\end{array}$

Now we consider the Lagrangian density of the electromagnetic field $(\mathbf{E},\mathbf{H})$

$\begin{array}{c}\hfill {\mathcal{L}}_0={\displaystyle \frac{1}{2}}\left[{\left|\mathbf{E}\right|}^2-{\left|\mathbf{H}\right|}^2\right]={\displaystyle \frac{1}{2}}|{\mathbf{A}}_t{+\nabla \varphi |}^2-{\displaystyle \frac{1}{2}}{|\nabla \times \mathbf{A}|}^2\end{array}$

and so the total action is given by

$\mathcal{S}=\int \int {\mathcal{L}}_{\mathcal{KGM}}+{\mathcal{L}}_0.$

Varying $\mathcal{S}$ with respect to $\delta u$, $\delta S$, $\delta \varphi$ and $\delta A$, respectively, we get

(4)$\begin{array}{c}\hfill \square u+\left[{|\nabla S-\mathrm{e}\mathbf{A}|}^2-{\left({\displaystyle \frac{\partial S}{\partial t}}+\mathrm{e}\varphi \right)}^2+m^2\right]u-{\left|u\right|}^{p-2}u=0,\end{array}$

(5)$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}\left[\left({\displaystyle \frac{\partial S}{\partial t}}+\mathrm{e}\varphi \right)u^2\right]-\nabla ·\left[(\nabla S-\mathrm{e}\mathbf{A})u^2\right]=0,\end{array}$

(6)$\begin{array}{c}\hfill \nabla ·\left({\displaystyle \frac{\partial \mathbf{A}}{\partial t}}+\nabla \varphi \right)=\mathrm{e}\left({\displaystyle \frac{\partial S}{\partial t}}+\mathrm{e}\varphi \right)u^2,\end{array}$

(7)$\begin{array}{c}\hfill \nabla \times (\nabla \times \mathbf{A})+{\displaystyle \frac{\partial }{\partial t}}\left({\displaystyle \frac{\partial \mathbf{A}}{\partial t}}+\nabla \varphi \right)=\mathrm{e}\left(\nabla S-\mathrm{e}\mathbf{A}\right)u^2.\end{array}$

In order to obtain the standing waves, let

$u=u\left(x\right),S=\omega t,\mathbf{A}=0,\varphi =\varphi \left(x\right),$

With this ansatz, Equations (5) and (7) are identically satisfied, while (4) and (6) become

(8)$-\Delta u+\left[m^2-{(\omega +\mathrm{e}\varphi )}^2\right]u-{\left|u\right|}^{p-2}u=0,$

(9)$-\Delta \varphi =-\mathrm{e}(\omega +\mathrm{e}\varphi )u^2.$

Since ${\mathrm{e}}^2=1$, we take $\mathrm{e}=1$; then (8) and (9) become

(10)$-\Delta u+\left[m^2-{(\omega +\varphi )}^2\right]u-{\left|u\right|}^{p-2}u=0,$

(11)$-\Delta \varphi =-(\omega +\varphi )u^2.$

The system composed of Equations (10) and (11) represents the form of the Klein–Gordon–Maxwell system to be discussed next.

This paper first introduces the variational framework for the Klein–Gordon–Maxwell system. Due to the strongly indefinite nature of the energy functional corresponding to Equations (10) and (11), critical point theory cannot be directly applied. To overcome this difficulty, a reduction method will be employed. However, by (11), the nonlocal term $\varphi$ cannot be explicitly expressed through u when regarding $\varphi$ as a function of u. This brings many difficulties to the research on the Klein–Gordon–Maxwell system. To help readers to better understand the nonlocal term $\varphi$, we will systematically explain the properties of $\varphi$ in Section 2.

Next, this paper will primarily explore the existence and nonexistence of solutions for the Klein–Gordon–Maxwell system under both autonomous and non-autonomous cases. Finally, we will investigate the solutions of the Klein–Gordon–Maxwell system in ${\mathbb{R}}^2$. Each case is further subdivided based on distinct mathematical structures, demonstrating the flexible application of variational methods and critical point theory in diverse situation.

First, we focus on the autonomous case in Section 3. Based on the growth properties of nonlinear terms, the relevant literature is classified into subcritical growth and critical growth cases. For subcritical nonlinear terms, combined with properties of $\varphi$ and techniques like the Pohozaev identity, we progressively expand the exponent range p in the nonlinearity ${\left|u\right|}^{p-2}u$ by analyzing the geometric structure of the energy functional. Furthermore, replacing ${\left|u\right|}^{p-2}u$ with general nonlinearities $f\left(u\right)$, the existence of ground state solutions and the multiplicity of solutions for the Klein–Gordon–Maxwell system are also investigated under both the classical Ambrosetti–Rabinowitz condition and Berestycki–Lions-type conditions. Subsequently, the existence and nonexistence of solutions under critical growth conditions are discussed. The lack of compactness induced by critical exponents further complicates the study of solution existence. By estimating the range of minimax levels associated with the energy functional, compactness is partially recovered and then the existence and nonexistence of solutions for the system can be established. Research on the critical exponent case includes the fractional Klein–Gordon–Maxwell system. At the end of this section, we consider the existence of solutions for the Klein–Gordon–Maxwell system under the zero-mass case, i.e., the limit case $m=\omega$.

In non-autonomous cases, potentials are further classified into coercive potentials, steep potential wells, periodic potentials, vanishing potentials and radial potentials based on their distinct properties in Section 4. For coercive potentials, this study primarily focuses on studying the existence of solutions under various subcritical growth conditions. The nonlinear term f satisfies a series of super-cubic growth conditions and superlinear growth conditions, which are listed in the text in order of progressively weaker strength. Moreover, research on sign-changing solutions for the Klein–Gordon–Maxwell system has so far yielded results only under coercive potentials. A steep potential well is more challenging to handle than coercive potentials. Unlike coercive potentials, the energy functional under a steep potential well no longer satisfies compactness conditions. Research on steep potential wells is divided into subcritical and critical cases based on the nonlinear terms and studies both the existence and asymptotic behavior of solutions. Notably, this section introduces the semiclassical Klein–Gordon–Maxwell system. Due to the mathematical structure of semiclassical frameworks, researchers subtly apply scaling transformations to reformulate the problem into a steep potential well setting. Under periodic potentials, for subcritical nonlinear terms, studies primarily focus on how the existence of system solutions evolves as the conditions for the nonlinear term f are progressively relaxed. In critical cases, the emphasis shifts to optimizing results by adjusting the range of the infimum associated with periodic potentials, achieving improved outcomes as the range expands. Regarding vanishing potentials, the existing literature remains limited and mainly addresses subcritical scenarios. Finally, the existence of solutions under radial potentials are investigated.

The subsequent discussion focuses on the study of solutions for the Klein–Gordon–Maxwell system in ${\mathbb{R}}^2$ in Section 5. So far, only three papers have explored the existence of solutions in this setting. Unlike the case in ${\mathbb{R}}^3$, the critical exponent in ${\mathbb{R}}^2$ shifts from $2^{*}$ to ∞, and $H^1\left({\mathbb{R}}^2\right)$ is no longer continuously embedded into $L^{\infty }\left({\mathbb{R}}^2\right)$. These investigations significantly expand the applicability of variational methods in this domain.

In summary, this paper aims to synthesize and analyze research progress on solving various problems in autonomous and non-autonomous Klein–Gordon–Maxwell systems using variational methods through a literature review, providing researchers in the field with a clear research framework and insights for future directions. For reader convenience, results from the cited references are accompanied by concise outlines of their proofs; detailed arguments can be found in the original sources.

2. Variational Framework for the Klein–Gordon–Maxwell System

Consider the functional

(12)$\begin{array}{c}\hfill F(u,\varphi )={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^3}\left\{{|\nabla u|}^2-{|\nabla \varphi |}^2+[m^2-{(\omega +\varphi )}^2]u^2\right\}dx-{\displaystyle \frac{1}{p}}{\int }_{{\mathbb{R}}^3}{\left|u\right|}^{p}dx.\end{array}$

The following proposition holds.

Lemma 1

([1]). F is ${\mathcal{C}}^1$ on $H^1\left({\mathbb{R}}^3\right)\times D^{1,2}\left({\mathbb{R}}^3\right)$ and its critical points are the solutions of (10) and (11).

The functional F is neither bounded from below nor from above and this indefiniteness cannot be removed by a compact perturbation. For this reason, the usual tools of critical point theory cannot be used in a direct way. To avoid this difficulty, we reduce the study of (12) to a functional of the only variable u.

Lemma 2

([1]). For any fixed $u\in H^1\left({\mathbb{R}}^3\right)$, there exists a unique $\varphi ={\varphi }_u\in D^{1,2}\left({\mathbb{R}}^3\right)$, which solves the equation

$\begin{array}{c}\hfill -\Delta \varphi =-(\omega +\varphi )u^2.\end{array}$

By Lemma 2, we can define $\mathrm{\Phi }:H^1\left({\mathbb{R}}^3\right)\to D^{1,2}\left({\mathbb{R}}^3\right)$, such that for all $u\in H^1\left({\mathbb{R}}^3\right)$, $\mathrm{\Phi }\left(u\right)$ is the unique solution of (11), namely $\mathrm{\Phi }\left(u\right)=-\omega L_{u^2}^{-1}\left(u^2\right).$ Also,

Lemma 3

([1]). The map Φ is ${\mathcal{C}}^1$ and its graph $G_{\mathrm{\Phi }}$ is given by

$G_{\mathrm{\Phi }}=\left\{(u,\varphi )\in H^1\left({\mathbb{R}}^3\right)\times D^{1,2}\left({\mathbb{R}}^3\right):F_{\varphi }^{\prime }(u,\varphi )=0\right\}.$

For any fixed $u\in H^1\left({\mathbb{R}}^3\right)$, ${\varphi }_u$ solves (11), and then

(13)$\begin{array}{c}\hfill -\Delta {\varphi }_u+u^2{\varphi }_u=-\omega u^2.\end{array}$

Multiplying by ${\varphi }_u$ and integrating by parts leads to

(14)$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^3}{\left|\nabla {\varphi }_u\right|}^{2}dx+{\int }_{{\mathbb{R}}^3}{u}^2{\varphi }_u^{2}dx=-\omega {\int }_{{\mathbb{R}}^3}{u}^2{\varphi }_{u}dx.\end{array}$

Let

$I\left(u\right)=F(u,\varphi ).$

By Lemmas 1 and 3, I is ${\mathcal{C}}^1$. Substituting (14) into (12) yields

$\begin{array}{c}\hfill I\left(u\right)={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^3}\left({|\nabla u|}^2+(m^2-{\omega }^2)u^2\right)dx-{\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^3}\omega {\varphi }_u{u}^{2}dx-{\displaystyle \frac{1}{p}}{\int }_{{\mathbb{R}}^3}{\left|u\right|}^{p}dx.\end{array}$

The critical points of functionals F and I satisfy the following proposition.

Lemma 4

([1]). For any $(u,\varphi )\in H^1\left({\mathbb{R}}^3\right)\times D^{1,2}\left({\mathbb{R}}^3\right)$, the following statements are equivalent: (a) 

$(u,\varphi )$ is a critical point of F;

(b) 

u is a critical point of I and $\varphi ={\varphi }_u$.

Equation (11) dictates that ${\varphi }_u$ cannot be expressed explicitly as a function of u. In order to better investigate the existence of solutions for the Klein–Gordon–Maxwell system, we present the following lemmas that illuminate the properties of ${\varphi }_u$.

Lemma 5

([2]). For any $u\in H^1\left({\mathbb{R}}^3\right)$, one sees (a) 

$\parallel {\varphi }_u{\parallel }_{D^{1,2}\left({\mathbb{R}}^3\right)}\le {C\parallel u\parallel }_{L^{{\displaystyle \frac{12}{5}}}\left({\mathbb{R}}^3\right)}^2\le C{\parallel u\parallel }_{H^1\left({\mathbb{R}}^3\right)}^2$;

(b) 

${\int }_{{\mathbb{R}}^3}{\varphi }_u{u}^{2}dx\le {C\parallel u\parallel }_{L^{{\displaystyle \frac{12}{5}}}\left({\mathbb{R}}^3\right)}^4\le C{\parallel u\parallel }_{H^1\left({\mathbb{R}}^3\right)}^4$.

Lemma 6

([3]). For any $u\in H^1\left({\mathbb{R}}^3\right)$, $-\omega \le {\varphi }_u\le 0.$

Proof. 

For any $u\in H^1\left({\mathbb{R}}^3\right)$, by multiplying (13) by ${(\omega +{\varphi }_u)}_{-}:=-min\left\{\omega +{\varphi }_u,0\right\}$, which is an admissible test function, we get

$\begin{array}{c}\hfill {\int }_{{\varphi }_u<-\omega }{\left|\nabla {\varphi }_u\right|}^{2}dx+{\int }_{{\varphi }_u<-\omega }{\left(\omega +{\varphi }_u\right)}^2{u}^{2}dx=0,\end{array}$

which implies

${\varphi }_u\ge -\omega .$

Next we use ${\left({\varphi }_u\right)}_{+}:=max\left\{{\varphi }_u,0\right\}$ as a test function in (13) to get

$\begin{array}{c}\hfill {\int }_{{\varphi }_u\ge 0}{\left|\nabla {\varphi }_u\right|}^{2}dx+{\int }_{{\varphi }_u\ge 0}{\left(\omega +{\varphi }_u\right)}^2{u}^{2}dx=0,\end{array}$

which implies

${\varphi }_u\le 0.$

□

Lemma 7

([4]). For any $\alpha \in (2,3)$, there exist $\beta >\alpha$ and $C>0$ such that

${\int }_{{\mathbb{R}}^3}|{\varphi }_u|u^{2}dx\le C{\parallel u\parallel }_{L^{\alpha }\left({\mathbb{R}}^3\right)}^{\beta }.$

Lemma 8

([1]). If u is radially symmetric, then ${\varphi }_u$ is radially symmetric.

Since ${\varphi }_u$ cannot be expressed explicitly as a function of u, general scaling transformations are no longer applicable to ${\varphi }_u$, except for the following special case.

Lemma 9([5]). Let $u_{\lambda }\left(x\right):=u\left(\lambda x\right),\lambda \in {\mathbb{R}}^{+}$; then $\mathrm{\Phi }\left(\lambda u_{\lambda }\right)={\left(\mathrm{\Phi }\left[u\right]\right)}_{\lambda }.$

Indeed,

$\begin{array}{cc}\hfill \Delta \left(\mathrm{\Phi }\left[u\right]\left(\lambda x\right)\right)& ={\lambda }^2\Delta \mathrm{\Phi }\left[u\right]\left(\lambda x\right)\hfill \\ \hfill & ={\lambda }^2\left(u^2\left(\lambda x\right)\mathrm{\Phi }\left[u\right]\left(\lambda x\right)+\omega u^2\left(\lambda x\right)\right)\hfill \\ \hfill & ={\left(\lambda u\left(\lambda x\right)\right)}^2\mathrm{\Phi }\left[u\right]\left(\lambda x\right)+\omega {\left(\lambda u\left(\lambda x\right)\right)}^2,\hfill \end{array}$

Combined with the uniqueness of ${\varphi }_u$ in Lemma 2, we complete the proof of Lemma 9.

The next three lemmas will study the properties of the derivative for ${\varphi }_u$.

Lemma 10

([6]). The map $\mathrm{\Phi }:u\in H^1\left({\mathbb{R}}^3\right)\to {\varphi }_u\in D^{1,2}\left({\mathbb{R}}^3\right)$ is ${\mathcal{C}}^1$. And for any $u,v\in H^1\left({\mathbb{R}}^3\right)$, one can observe that

(15)$\begin{array}{c}\hfill {\mathrm{\Phi }}^{\prime }\left(u\right)v=2{(\Delta -u^2)}^{-1}\left((\omega +{\varphi }_u)uv\right).\end{array}$

By using (15), for any $u,v\in H^1\left({\mathbb{R}}^3\right)$, we get

$\begin{array}{cc}\hfill I^{\prime }\left(u\right)v=& {\int }_{{\mathbb{R}}^3}\nabla u\nabla vdx+(m^2-{\omega }^2){\int }_{{\mathbb{R}}^3}uvdx-\omega {\int }_{{\mathbb{R}}^3}uv{\varphi }_{u}dx\hfill \\ \hfill & -\omega {\int }_{{\mathbb{R}}^3}{u}^2{(\Delta -u^2)}^{-1}\left((\omega +{\varphi }_u)uv\right)dx-{\int }_{{\mathbb{R}}^3}{\left|u\right|}^{p-2}uvdx\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^3}\nabla u\nabla vdx+(m^2-{\omega }^2){\int }_{{\mathbb{R}}^3}uvdx-\omega {\int }_{{\mathbb{R}}^3}uv{\varphi }_{u}dx\hfill \\ \hfill & -{\int }_{{\mathbb{R}}^3}\left(\omega u^2\right){(\Delta -u^2)}^{-1}\left((\omega +{\varphi }_u)uv\right)dx-{\int }_{{\mathbb{R}}^3}{\left|u\right|}^{p-2}uvdx.\hfill \end{array}$

As ${(\Delta -u^2)}^{-1}\left(\omega u\right)={\varphi }_u$, its derivative functional is

(16)$\begin{array}{c}\hfill I^{\prime }\left(u\right)v={\int }_{{\mathbb{R}}^3}\nabla u\nabla vdx+(m^2-{\omega }^2){\int }_{{\mathbb{R}}^3}uvdx-{\int }_{{\mathbb{R}}^3}(2\omega +{\varphi }_u){\varphi }_{u}uvdx-{\int }_{{\mathbb{R}}^3}{\left|u\right|}^{p-2}uvdx.\end{array}$

Lemma 11

([6]). Let $u\in H^1\left({\mathbb{R}}^3\right)$, and set ${\psi }_u:={\displaystyle \frac{1}{2}}{\mathrm{\Phi }}^{\prime }\left(u\right)u\in D^{1,2}\left({\mathbb{R}}^3\right)$. Then the following hold. (a) 

${\psi }_u$ is a solution to the integral equation

(17)$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^3}\omega {\psi }_u{u}^{2}dx={\int }_{{\mathbb{R}}^3}(\omega +{\varphi }_u){\varphi }_u{u}^{2}dx\end{array}$

and satisfies

$\Delta {\psi }_u=\left({\psi }_u+\omega +{\varphi }_u\right)u^2.$

(b) 

For almost everywhere on the set $\{x\in {\mathbb{R}}^3:u\left(x\right)\ne 0\}$,

$max\{-\omega -{\varphi }_u,{\varphi }_u\}\le {\psi }_u\le 0.$

Remark 1

([6]). By using Lemma 11, $-\omega \le {\psi }_u\le 0$ on $\{x\in {\mathbb{R}}^3:u\left(x\right)\ne 0\}$.

Remark 2

([7]). Let

$\begin{array}{c}\hfill G\left(u\right):=⟨I^{\prime }\left(u\right),u⟩={\int }_{{\mathbb{R}}^3}\left({|\nabla u|}^2+(m^2-{\omega }^2)u^2-(2\omega +{\varphi }_u){\varphi }_u{u}^2\right)dx-{\int }_{{\mathbb{R}}^3}{\left|u\right|}^{p}dx.\end{array}$

It follows from Lemma 11 that

$\begin{array}{cc}\hfill ⟨G^{\prime }\left(u\right),u⟩=& 2{\int }_{{\mathbb{R}}^3}\left({|\nabla u|}^2+(m^2-{\omega }^2)u^2\right)dx-8{\int }_{{\mathbb{R}}^3}\omega {\varphi }_u{u}^{2}dx\hfill \\ \hfill & -6{\int }_{{\mathbb{R}}^3}{\varphi }_u^2{u}^{2}dx-4{\int }_{{\mathbb{R}}^3}{\varphi }_u{\psi }_{u}dx-p{\int }_{{\mathbb{R}}^3}{\left|u\right|}^{p}dx.\hfill \end{array}$

Lemma 12

([7]). If $u_n\rightharpoonup u_0$ in $H^1\left({\mathbb{R}}^3\right)$ then, up to subsequences, ${\varphi }_{u_n}\rightharpoonup {\varphi }_{u_0}$ in $D^{1,2}\left({\mathbb{R}}^3\right)$. As a consequence $I^{\prime }\left(u_n\right)\to I^{\prime }\left(u_0\right)$ in the sense of distributions.

Remark 3

([8]). If $u_n\rightharpoonup u_0$ in $H_r^1\left({\mathbb{R}}^3\right)$, then ${\varphi }_{u_n}\to {\varphi }_{u_0}$ in $D_r^{1,2}\left({\mathbb{R}}^3\right)$.

3. The Existence of Solutions for the Autonomous Klein–Gordon–Maxwell System

In this chapter, we consider the following Klein–Gordon–Maxwell system:

(18)$\left\{\begin{array}{cc}-\Delta u+\left(m^2-{\omega }^2\right)u-\left(2\omega +\varphi \right)\varphi u={\left|u\right|}^{p-2}u,\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3,\hfill \end{array}\right.$

where $m>\omega >0$ are constants and $u,\varphi$ are unknowns.

3.1. The Existence and Non-Existence of Solutions for the Klein–Gordon–Maxwell System Under the Subcritical Growth Condition

Benci and Fortunato in [1] first studied the Klein–Gordon–Maxwell system by using the variational method. They obtained infinitely many radially symmetric solutions for system (18) when $4<p<6$. Later, based on the property $-\omega \le \varphi \le 0$ in Lemma 6, D’Aprile and Mugnai in [3] improved the result provided that $2\left({\displaystyle \frac{{\omega }^2}{m^2}}+1\right)<p<6$. If $u\in H^1\left({\mathbb{R}}^3\right)$ is a solution of system (18), then u satisfies the following Pohozaev identity:

(19)$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^3}\left({|\nabla u|}^2+(m^2-{\omega }^2)u^2+5\omega {\varphi }_u{u}^2-2{\varphi }_u^2{u}^2\right)dx-{\displaystyle \frac{6}{p}}{\int }_{{\mathbb{R}}^3}{\left|u\right|}^{p}dx=0.\end{array}$

With the aid of the Pohozaev identity and monotonicity trick (see [9]), Azzollini and Pomponi in [10] proved the existence of a nontrivial solution for system (18) if $3-\sqrt{1-{\displaystyle \frac{{\omega }^2}{m^2}}}<p<6$. Obviously,

$3-\sqrt{1-{\displaystyle \frac{{\omega }^2}{m^2}}}<2\left({\displaystyle \frac{{\omega }^2}{m^2}}+1\right)<4.$

In [7], the existence of ground state solutions for system (18) was first studied by using the Nehari manifold if $2+{\displaystyle \frac{4{\omega }^2}{m^2+{\omega }^2}}<p<6$. Wang in [6] discussed a new property for ${\varphi }_u$, i.e.,

$max\{-\omega -{\varphi }_u,{\varphi }_u\}\le {\psi }_u\le 0,$

where ${\psi }_u:={\displaystyle \frac{1}{2}}{\mathrm{\Phi }}^{\prime }\left(u\right)u$. Combined with such a property, the range of p in [6] is further extended to ${\displaystyle \frac{2\left(m^2+{\omega }^2-\sqrt{m^4-{\omega }^4}\right)}{{\omega }^2}}<p<6$. Chen and Tang in [11] investigated the existence of ground state solutions by employing a method based on their definition, namely proving that there exists u attained the ground state energy m, where

$m:=\underset{u\in \mathcal{M}}{inf}I\left(u\right),\mathcal{M}:=\left\{u\in H^1\left({\mathbb{R}}^3\right)\setminus \left\{0\right\}:I^{\prime }\left(u\right)=0\right\}.$

The paper [11] obtained the existence of ground state solutions for system (18) if ${\displaystyle \frac{4m^2-4m\sqrt{m^2-{\omega }^2}}{{\omega }^2}}<p<6$. By using a special Palais–Smale sequence associated with the Pohozaev identity (shortened to (PPS) sequence for brevity), Chen and Tang in [12] established the ground states provided that $3-\sqrt{1-{\displaystyle \frac{{\omega }^2}{m^2}}}<p<6$. Clearly,

$3-\sqrt{1-{\displaystyle \frac{{\omega }^2}{m^2}}}<{\displaystyle \frac{4m^2-4m\sqrt{m^2-{\omega }^2}}{{\omega }^2}}<{\displaystyle \frac{2\left(m^2+{\omega }^2-\sqrt{m^4-{\omega }^4}\right)}{{\omega }^2}}<2+{\displaystyle \frac{4{\omega }^2}{m^2+{\omega }^2}}.$

Also, the paper [12] generalized the nonlinear term ${\left|u\right|}^{p-2}u$ in system (18) to the general term $f\left(u\right)$, namely

(20)$\left\{\begin{array}{cc}-\Delta u+\left(m^2-{\omega }^2\right)u-\left(2\omega +\varphi \right)\varphi u=f\left(u\right),\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3.\hfill \end{array}\right.$

The authors in [12] proved that system (20) has a ground state solution if $3-\sqrt{1-{\displaystyle \frac{{\omega }^2}{m^2}}}<\mu <6$, where $f\in \mathcal{C}(\mathbb{R},\mathbb{R})$ satisfies

(SC) 

for any $t\in \mathbb{R}$, there exists $2<p<6$ such that $\left|f\left(t\right)\right|\le C(1+|t{|}^{p-1})$; $f\left(t\right)=o\left(t\right)$ as $t\to 0$.

(AR) 

for any $t\in \mathbb{R}$, there exists $\mu >2$ such that $tf\left(t\right)\ge \mu F\left(t\right),$ where $F\left(t\right)={\int }_0^{t}f\left(s\right)dt$.

As demonstrated in [12], for any $\omega$, system (20) has a ground state solution, which is denoted as $u_{\omega }$. Notice that $-\omega \le {\varphi }_u\le 0$. When $\omega =0$, system (20) reduces to

(21)$-\Delta u+m^{2}u=f\left(u\right),x\in {\mathbb{R}}^3.$

Then, Liu and Tang in [13] first investigated the relationship of the ground state solutions between the Klein–Gordon–Maxwell system and Schrödinger equation, that is, $u_{\omega }\rightharpoonup u_0$ in $H^1\left({\mathbb{R}}^3\right)$, where $u_0$ is the ground state of Equation (21). The paper [13] obtained the existence of infinitely many solutions for system (20) simultaneously.

In 1983, Berestycki and Lions in [14,15] studied the following Schrödinger equation:

$-\Delta u=g\left(u\right),x\in {\mathbb{R}}^N,$

where $N\ge 3$, $g\in \mathcal{C}(\mathbb{R},\mathbb{R})$ is odd and satisfies (g₁) 

$-\infty <{lim\; inf}_{s\to 0^{+}}{\displaystyle \frac{g\left(s\right)}{s}}\le {lim\; sup}_{s\to 0^{+}}{\displaystyle \frac{g\left(s\right)}{s}}=-m<0$;

(g₂) 

$-\infty \le {lim\; sup}_{s\to +\infty }{\displaystyle \frac{g\left(s\right)}{s^{2^{*}-1}}}\le 0$;

(g₃) 

there exists $\zeta >0$ such that $G\left(\zeta \right)={\int }_0^{\zeta }g\left(s\right)ds>0$.

Such conditions are called Berestycki–Lions conditions (shortened to (BL) conditions for brevity). The ground state solution and infinitely many solutions are gained in [14,15]. In view of [14,15], $\left(g_1\right)$ is almost necessary while $\left(g_2\right)$ and $\left(g_3\right)$ are necessary for the existence of a nontrivial solution. Liu, Li and Tang in [4] investigated the existence and multiplicity of solutions for the following Klein–Gordon–Maxwell system under (BL) conditions, as well as the decay properties and asymptotic behavior of the solutions.

(22)$\left\{\begin{array}{c}-\Delta u-(2\omega +\varphi )\varphi u=g\left(u\right),x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,x\in {\mathbb{R}}^3.\hfill \end{array}\right.$

The main results can be stated as follows.

Theorem 1.

Assume that $\left(g_1\right)-\left(g_3\right)$ hold. Then there exists ${\omega }_0>0$ such that system (22) has a nontrivial positive solution for $0<\omega <{\omega }_0$; if g is also odd, then for any fixed $k\in \mathbb{N}$, there exists ${\omega }_k>0$ such that system (22) has at least k pairs $\pm u_1,\pm u_2,\dots ,\pm u_k$ solutions for $0<\omega <{\omega }_k$.

Proof. 

First, we prove a new property of ${\varphi }_u$; namely, for any $\alpha \in (2,3)$, there exist $\beta >\alpha$ and $C>0$ such that

${\int }_{{\mathbb{R}}^3}|{\varphi }_u|u^{2}dx\le C{\parallel u\parallel }_{\alpha }^{\beta }.$

Define a cut-off functional

$\begin{array}{cc}\hfill I^T\left(u\right)=& {\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^3}{|\nabla u|}^2-{\displaystyle \frac{1}{2}}\chi \left({\displaystyle \frac{{\parallel u\parallel }_{\alpha }^{\beta }}{T^{\beta }}}\right){\int }_{{\mathbb{R}}^3}\omega {\varphi }_u{u}^{2}dx-{\int }_{{\mathbb{R}}^3}G\left(u\right)dx,\hfill \end{array}$

where ${\parallel u\parallel }_{\alpha }^{\beta }:={\parallel u\parallel }_{L^{\alpha }\left({\mathbb{R}}^3\right)}^{\beta }$, and a cut-off function $\chi \in {\mathcal{C}}^{\infty }({\mathbb{R}}_{+},\mathbb{R})$, which is

$\begin{array}{c}\hfill \chi \left(s\right)=1,s\in [0,1];0\le \chi \left(s\right)\le 1,s\in (1,2);\chi \left(s\right)=0,s\in {[2,+\infty );\parallel \chi \parallel }_{\infty }^{\prime }\le 2.\end{array}$

Evidently, $I^T$ satisfies the mountain pass structure. From Lemma 9, one has

${\varphi }_{e^{\theta }u\left(e^{\theta }x\right)}\left(e^{\theta }x\right)={\varphi }_{u\left(e^{\theta }x\right)}\left(e^{\theta }x\right),\theta \in \mathbb{R},u\in H^1\left({\mathbb{R}}^3\right).$

Define an auxiliary functional

$\begin{array}{cc}\hfill \widetilde{I^T}(\theta ,u):=& I^T\left(\phi (\theta ,u)\right)\hfill \\ \hfill =& {\displaystyle \frac{e^{\theta }}{2}}{\int }_{{\mathbb{R}}^3}{|\nabla u|}^{2}dx-{\displaystyle \frac{\omega }{2e^{\theta }}}\chi \left({\displaystyle \frac{e^{\frac{\theta \beta }{\alpha }(\alpha -3)}{\parallel u\parallel }_{\alpha }^{\beta }}{T^{\beta }}}\right){\int }_{{\mathbb{R}}^3}{\varphi }_u{u}^{2}dx-{\displaystyle \frac{1}{e^{3\theta }}}{\int }_{{\mathbb{R}}^3}G\left(e^{\theta }u\right)dx.\hfill \end{array}$

$\widetilde{I^T}(\theta ,u)$ also satisfies the mountain pass structure. By using a quantitative deformation lemma (see [16]) for $\widetilde{I^T}$, one can obtain a (PPS) sequence $\left\{u_n\right\}$ as follows:

(23)$\begin{array}{c}\hfill I^T\left(u_n\right)\to c^T,{\left(I^T\right)}^{\prime }\left(u_n\right)\to 0,\mathcal{P}\left(u_n\right)\to 0,n\to \infty ,\end{array}$

where $\mathcal{P}\left(u\right)=0$ is the Pohozaev identity associated with system (20) and $c^T$ is the mountain pass energy level for the functional $\widetilde{I^T}$. Combined with the definition of the cut-off function $\chi$, one can easily see that a sequence satisfying (23) is bounded. Then there exists $u\in H_r^1\left({\mathbb{R}}^3\right)$ such that $u_n\to u$ in $H_r^1\left({\mathbb{R}}^3\right)$. It follows from $\left(g_1\right)-\left(g_3\right)$ and the definition of $\chi$ that

$\parallel u_{{\lambda }_n}{\parallel }^2\le C+C\omega T^{\beta }+C{\left(C_1+C_2\omega T^{\beta }\right)}^3.$

Then there exists T large enough and ${\omega }_0>0$ small enough such that

${\parallel u\parallel }_{\alpha }\le C\parallel u\parallel \le T\mathrm{for}0<\omega <{\omega }_0.$

Hence $\chi \left({\displaystyle \frac{{\parallel u\parallel }_{\alpha }^{\beta }}{T^{\beta }}}\right)=1$, which implies u is a nontrivial solution for system (22). Combined with a strong maximum principle, u is a positive solution. Similarly, the existence of multiple solutions can be established. □

Note that the paper [15] gained the existence of infinitely many solutions under (BL) conditions. However, due to the truncation of the functional, the paper [4] can only yield the existence of multiple pairs of solutions.

At last, D’Aprile and Mugnai in [5] studied the nonexistence of solutions for system (20). They proved that there are only trivial solutions for system (20) if one of the following conditions holds:(i) 

$f\left(s\right)s+2(m^2-{\omega }^2)s^2\ge 6F\left(s\right)$, $\forall s\in \mathbb{R}$;

(ii) 

$F\left(s\right)s\ge f\left(s\right)s$, $\forall s\in \mathbb{R}$.

3.2. The Existence and Non-Existence of Solutions for the Klein–Gordon–Maxwell System Under the Critical Growth Condition

Consider the following Klein–Gordon–Maxwell system:

(24)$\left\{\begin{array}{cc}-\Delta u+\left(m^2-{\omega }^2\right)u-\left(2\omega +\varphi \right)\varphi u={\theta \left|u\right|}^{q-2}u+{\left|u\right|}^{2^{*}-2}u,\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3.\hfill \end{array}\right.$

Cassani in [8] first discussed the existence of solutions for the Klein–Gordon–Maxwell system under critical growth conditions. With the method of Brezis–Nirenberg (see [17]), the paper [8] proved that system (24) has at least a radially symmetric nontrivial solution if $4<q<6$. Moreover, system (24) possesses a nontrivial solution provided that $q=4$ and $\theta$ is sufficiently large. Meanwhile, system (24) has only a trivial solution if $\theta =0$. Carriao in [18] further investigated the case $2<q<4$. They found that system (24) has at least a radially symmetric nontrivial solution if $m\sqrt{q-2}>\omega \sqrt{2}$ and $\theta$ is sufficiently large. By refining the above result through the monotonicity trick, Wang in [19] established the existence of a nontrivial radial solution for system (24), if one of the following conditions holds:(i) 

$4<q<6,\theta >0$;

(ii) 

$3<q\le 4$, $\theta$ sufficiently large;

(iii) 

$2<q\le 3$, $m\sqrt{(q-2)(4-q)}>\omega$, $\theta$ sufficiently large.

Afterwards, Tang, Wen and Chen in [20] generalized the nonlinear term ${\left|u\right|}^{p-2}u$ in system (24) to the general term $g\left(u\right)$, namely

(25)$\left\{\begin{array}{cc}-\Delta u+\left(m^2-{\omega }^2\right)u-\left(2\omega +\varphi \right)\varphi u=\theta g\left(u\right)+{\left|u\right|}^{2^{*}-2}u,\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3,\hfill \end{array}\right.$

where g satisfies $\left(SC\right),\left(AR\right)$ and

(G) 

$G\left(t\right)\ge 0$ for $t\ge 0$; if $\mu \in (2,4]$, there exist $\alpha >0$ and $p\in (2,4]$ such that $G\left(t\right)\ge \alpha t^p$ for all $t\ge 1$.

The main results can be obtained as follows.

Theorem 2.

System (25) has a ground state solution if one of the following conditions holds: (i) 

$4<\mu <6,\theta >0$;

(ii) 

$3<\mu \le 4$, θ sufficiently large;

(iii) 

$2<\mu \le 3$, $0<\omega <m\sqrt{(\mu -2)(4-\mu )}$, θ sufficiently large.

Proof. 

The corresponding energy functional of system (25) is

$\begin{array}{c}\hfill I\left(u\right)={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^3}\left({|\nabla u|}^2+(m^2-{\omega }^2)u^2\right)dx-{\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^3}\omega {\varphi }_u{u}^{2}dx-{\int }_{{\mathbb{R}}^3}\left(\theta G\left(u\right)+{\displaystyle \frac{1}{2^{*}}}{\left|u\right|}^{2^{*}}\right)dx.\end{array}$

Clearly, I satisfies the mountain pass structure. The Pohozaev identity corresponding to the critical points of the functional I can be written as

$\begin{array}{c}\hfill \mathcal{P}\left(u\right)={\int }_{{\mathbb{R}}^3}\left({|\nabla u|}^2+(m^2-{\omega }^2)u^2+5\omega {\varphi }_u{u}^2-2{\varphi }_u^2{u}^2\right)dx-6\theta {\int }_{{\mathbb{R}}^3}G\left(u\right)dx-{\displaystyle \frac{6}{2^{*}}}{\int }_{{\mathbb{R}}^3}{\left|u\right|}^{2^{*}}dx=0.\end{array}$

Similar to the proof of (23) in Theorem 1, a (PPS) sequence $\left\{u_n\right\}$ corresponding to I can be obtained as follows:

(26)$\begin{array}{c}\hfill I\left(u_n\right)\to c,I^{\prime }\left(u_n\right)\to 0,\mathcal{P}\left(u_n\right)\to 0,n\to \infty ,\end{array}$

where c is the mountain pass energy level for the functional I. It follows from the property $-\omega \le {\varphi }_u\le 0$ of the nonlocal term that

(27)$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^3}\omega {\varphi }_u{u}^{2}dx\le {\omega }^2{\parallel u\parallel }_{L^2\left({\mathbb{R}}^3\right)}^2.\end{array}$

Combined with the method of Brezis–Nirenberg and (27), one can see that

$\begin{array}{c}\hfill 0<c<{\displaystyle \frac{1}{3}}{S}^{{\displaystyle \frac{3}{2}}}.\end{array}$

Then,

$\begin{array}{c}\hfill \underset{n\to \infty }{lim\; sup}\underset{y\in {\mathbb{R}}^3}{sup}{\int }_{B_1\left(y\right)}{u}_n^{2}dx>0,\end{array}$

where $\left\{u_n\right\}$ satisfies (26). It is easy to see that $\left\{u_n\right\}$ is bounded, and then there exists $u\in H^1\left({\mathbb{R}}^3\right)$ such that $u_n\rightharpoonup u\ne 0$. Define the ground state energy and solution set

$m:=\underset{u\in \mathcal{M}}{inf}I\left(u\right),\mathcal{M}:=\{u\in H^1\left({\mathbb{R}}^3\right)\setminus \left\{0\right\}:I^{\prime }\left(u\right)=0\}.$

As discussed above, it follows that $\mathcal{M}\ne \varnothing$. Let $\left\{u_n\right\}\subset \mathcal{M}$ with $I\left(u_n\right)\to m$. By repeating the preceding proof, we derive that $\left\{u_n\right\}$ is bounded. Hence, there is $\widehat{u}\in H^1\left({\mathbb{R}}^3\right)\setminus \left\{0\right\}$ such that $I^{\prime }\left(\widehat{u}\right)=0$ and $I\left(\widehat{u}\right)\ge m$. From Fatou’s Lemma and conditions $\left(i\right)-\left(iii\right)$, $I\left(\widehat{u}\right)=m$. □

Note that unlike previous works, in Theorem 2, the authors established a specific value for $\theta$; i.e., they explicitly computed a threshold ${\theta }^{*}$ such that Theorem 2 holds for $\theta >{\theta }^{*}$. For more details, the readers are referred to [20].

Zhang in [21] generalized the conditions and conclusions of [8] to the following fractional Klein–Gordon–Maxwell system:

$\left\{\begin{array}{cc}{(-\Delta )}^{s}u+\left(m^2-{\omega }^2\right)u-\left(2\omega +\varphi \right)\varphi u={\theta \left|u\right|}^{q-2}u+{\left|u\right|}^{2^{*}-2}u,\hfill & x\in {\mathbb{R}}^3,\hfill \\ {(-\Delta )}^s\varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3.\hfill \end{array}\right.$

where

$\begin{array}{c}\hfill {(-\Delta )}^{s}u\left(x\right)=C_{N,s}PV{\int }_{{\mathbb{R}}^3}{\displaystyle \frac{u\left(x\right)-u\left(y\right)}{{|x-y|}^{3+2s}}}dy.\end{array}$

3.3. The Existence of Solutions for the Klein–Gordon–Maxwell System Under the Zero-Mass Case

In this section, we consider the limit case $m=\omega$. Azzollini and Pomponi in [10] first studied the zero-mass case, namely

(28)$\left\{\begin{array}{cc}-\Delta u-\left(2\omega +\varphi \right)\varphi u=f\left(u\right),\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3,\hfill \end{array}\right.$

where f satisfies (Q₁) 

for all $t\in \mathbb{R}$, there exist $4<p_0\le p_1<6<p_2$ and $C_1,C_2>0$ such that

$f\left(t\right)t\ge p_{0}F\left(t\right),F\left(t\right)\ge C_1{min\left\{\right|t|}^{p_1}{,\left|t\right|}^{p_2}\},|f\left(t\right)|\le C_2{min\left\{\right|t|}^{p_1-1}{,\left|t\right|}^{p_2-1}\},$

and obtained a nontrivial solution. Azzollini in [22] considered the following Klein–Gordon–Maxwell system:

(29)$\left\{\begin{array}{cc}-\Delta u-\left(2\omega +\varphi \right)\varphi u=u^{p-1},\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3.\hfill \end{array}\right.$

For any $p\in (3,6)$, there exists a finite solution $(u,\varphi )\in {\mathcal{C}}^2\left({\mathbb{R}}^3\right)\times {\mathcal{C}}^2\left({\mathbb{R}}^3\right)$ for system (29). Moreover, u decays exponentially at infinity and there exist two positive constants $C_1,C_2$ such that ${\displaystyle \frac{C_1}{r}}\le {\varphi }_u\le {\displaystyle \frac{C_2}{r}}$ for $\left|R\right|\ge 1$. The paper [12] extends $\left(Q1\right)$ to the following condition:(Q₂) 

for all $t\in \mathbb{R}$, there exist $3<p_0\le p_1<6<p_2$ and $C_1,C_2>0$ such that

$f\left(t\right)t\ge p_{0}F\left(t\right),F\left(t\right)\ge C_1{min\left\{\right|t|}^{p_1}{,\left|t\right|}^{p_2}\},|f\left(t\right)|\le C_2{min\left\{\right|t|}^{p_1-1}{,\left|t\right|}^{p_2-1}\}.$

The main result can be obtained as follows.

Theorem 3.

System (28) has a nontrivial solution $(u,\varphi )\in {\mathcal{D}}^{1,2}\left({\mathbb{R}}^3\right)\times {\mathcal{D}}^{1,2}\left({\mathbb{R}}^3\right)$.

Proof. 

$Step1.$ We can find a weak limit of solution $(u_{\epsilon },{\varphi }_{\epsilon })\in {\mathcal{H}}^1\left({\mathbb{R}}^3\right)\times {\mathcal{D}}^{1,2}\left({\mathbb{R}}^3\right)$ for the following approximation problem:

$\left\{\begin{array}{cc}-\Delta u+\left[\epsilon -\left(2\omega +\varphi \right)\varphi \right]u=f\left(u\right),\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3.\hfill \end{array}\right.$

The solution $(u_{\epsilon },{\varphi }_{\epsilon })$ is the ground state such that $0<I_{\epsilon }\left(u_{\epsilon }\right)={inf}_{{\mathcal{M}}_{\epsilon }}{I}_{\epsilon }\le c_{\epsilon }$, where ${\mathcal{M}}_{\epsilon }:=\{u\in H^1\left({\mathbb{R}}^3\right)\setminus \left\{0\right\}:I_{\epsilon }^{\prime }\left(u\right)=0\}$ and $c_{\epsilon }$ is the mountain pass energy level for the functional $I_{\epsilon }$. And there exists a constant $C_0>0$ independent of $\epsilon$ such that $c_{\epsilon }\le C_0$ for all $\epsilon \le 1$. Moreover, there exists a constant $\overline{C}$ independent of $\epsilon$ such that

(30)$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^3}f\left(u_{\epsilon }\right)u_{\epsilon }dx\ge \overline{C}.\end{array}$

$Step2.$ Choose a sequence $\left\{{\epsilon }_n\right\}\subset (0,1]$ such that ${\epsilon }_n\to 0$. Then there exists a sequence $\left\{u_{{\epsilon }_n}\right\}\subset {\mathcal{M}}_{{\epsilon }_n}$ such that $0<I_{{\epsilon }_n}\left(u_{{\epsilon }_n}\right)={inf}_{{\mathcal{M}}_{{\epsilon }_n}}{I}_{{\epsilon }_n}\le c_{{\epsilon }_n}\le C_0$. Obviously, $\left\{u_{{\epsilon }_n}\right\}$ is a (PPS) sequence, together with the fact that

$\begin{array}{c}\hfill \parallel \nabla u_{{\epsilon }_n}{\parallel }_{L^2\left({\mathbb{R}}^3\right)}^2+\parallel \nabla {\varphi }_{u_{{\epsilon }_n}}{\parallel }_{L^2\left({\mathbb{R}}^3\right)}^2={\parallel \nabla u_{{\epsilon }_n}\parallel }_{L^2\left({\mathbb{R}}^3\right)}^2-{\int }_{{\mathbb{R}}^3}\left(\omega +{\varphi }_{u_{{\epsilon }_n}}\right){\varphi }_{u_{{\epsilon }_n}}{u}_{{\epsilon }_n}^{2}dx\le 2C_0+{\displaystyle \frac{3C_0}{p_0-3}},\end{array}$

Then $\left\{\left(u_{{\epsilon }_n},{\varphi }_{u_{{\epsilon }_n}}\right)\right\}$ is bounded in $D^{1,2}\left({\mathbb{R}}^3\right)\times {\mathcal{D}}^{1,2}\left({\mathbb{R}}^3\right)$. Similar to (30), there exists a nontrivial solution $\left(u_0,{\varphi }_{u_0}\right)\in {\mathcal{D}}^{1,2}\left({\mathbb{R}}^3\right)\times {\mathcal{D}}^{1,2}\left({\mathbb{R}}^3\right)$ for system (28). □

4. The Existence of Solutions for the Non-Autonomous Klein–Gordon–Maxwell System

In this chapter, we consider the following Klein–Gordon–Maxwell system:

(31)$\left\{\begin{array}{cc}-\Delta u+V\left(x\right)u-\left(2\omega +\varphi \right)\varphi u=f(x,u),\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3,\hfill \end{array}\right.$

where V is a potential function. We study the existence and asymptotic behavior of solutions for the Klein–Gordon–Maxwell system under different conditions imposed on the potential V.

4.1. The Existence of Solutions for the Klein–Gordon–Maxwell System with Coercive Potential

We call V the coercive potential if one of the following conditions holds:(V₁) 

$V\left(x\right)\to \infty$ as $\left|x\right|\to \infty$.

(V₂) 

for any $M>0$, $\mathrm{meas}\left(\{x\in {\mathbb{R}}^3:V\left(x\right)\le M\}\right)<\infty$.

(V₃) 

for any $M,r>0$,

$\mathrm{meas}\left(\{x\in B_r\left(y\right):V\left(x\right)\le M\}\right)\to 0,\left|y\right|\to \infty .$

(V₄) 

there exists ${\nu }_0>0$ such that

$\underset{\left|y\right|\to \infty }{lim}\mathrm{meas}\left(\{x\in {\mathbb{R}}^3:|x-y|\le {\nu }_0,V\left(x\right)\le M\}\right)=0\mathrm{for}\mathrm{any}M>0.$

He in [23] first investigated the existence of solutions for the Klein–Gordon–Maxwell system with coercive potential. In [23], the nonlinear term f is subcritical in whole space and superlinear at the origin, i.e., (SC)′ 

For any $x\in {\mathbb{R}}^3$, there exists $2<p<6$ such that $\left|f(x,t)\right|\le C(1+|t{|}^{p-1})$; $f(x,t)=o\left(t\right)$ as $t\to 0$.

Additionally, this is the case if f is odd and satisfies one of the following conditions:(F₁) 

for any $(x,t)\in {\mathbb{R}}^3\times \mathbb{R}$, there exists $\mu >4$ such that $tf(x,t)\ge \mu F(x,t)$;

(F₂) 

for any $x\in {\mathbb{R}}^3$, ${\displaystyle \frac{f(x,t)}{t^3}}\to \infty$ and $\tilde{F}(x,t):={\displaystyle \frac{1}{4}}f(x,t)t-F(x,t)\to \infty$ as $\left|s\right|\to \infty$.

The symmetric mountain pass geometry and boundedness of Palais–Smale sequence can be established. The author proved the existence of infinitely many solutions by using the symmetric mountain pass lemma. Ding et al. in [24] and Li et al. in [25] replace $\left(F_1\right)$ or $\left(F_2\right)$ with a weaker condition $\left(F_3\right)-\left(F_4\right)$, i.e., (F₃) 

for any $x\in {\mathbb{R}}^3$, there exist $\vartheta ,r_0>0$ such that $f(x,t)t-4F(x,t)+\vartheta t^2\ge 0$ for $\left|t\right|\ge r_0$;

(F₄) 

for any $x\in {\mathbb{R}}^3$, ${lim}_{\left|t\right|\to \infty }{\displaystyle \frac{F(x,t)}{t^4}}=\infty$.

Notice that all the above conditions correspond to the super-cubic growth of f. Chen and Song in [26] studied the case where f exhibits superlinear growth. When f satisfies ${\left(AR\right)}^{\prime }$, i.e., (AR)′ 

for any $(x,t)\in {\mathbb{R}}^3\times \mathbb{R}$, there exists $\mu >2$ such that $tf(x,t)\ge \mu F(x,t),$

the paper [26] is concerned with a kind of nonhomogeneous Klein–Gordon–Maxwell system

(32)$\left\{\begin{array}{cc}-\Delta u+V\left(x\right)u-\left(2\omega +\varphi \right)\varphi u=f(x,u)+h\left(x\right),\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3.\hfill \end{array}\right.$

By using Ekeland’s principle and the mountain pass lemma, the authors gained two nontrivial solutions. Chen and Tang in [11] weakened the condition ${\left(AR\right)}^{\prime }$ to (F₅) 

for all $(x,t)\in {\mathbb{R}}^3\times \mathbb{R}$, there exist $\mu >2$ and $\vartheta >0$ such that $f(x,t)t-\mu F(x,t)+\vartheta t^2\ge 0$;

(F₆) 

for all $x\in {\mathbb{R}}^3$, ${lim}_{\left|t\right|\to \infty }{\displaystyle \frac{F(x,t)}{t^2}}=+\infty$.

The paper [11] obtained the existence of infinitely many solutions for system (31) provided that f is also odd. Afterwards, Wang in [27] expanded the above conditions to the case where the nonlinearity is nonhomogeneous and gained two nontrivial solutions for system (32). Wu and Lin in [28] considered another case involving superlinear growth, where g satisfies (F₇) 

for all $(x,t)\in {\mathbb{R}}^3\times \mathbb{R}$, $f(x,t)t\ge 2F(x,t)\ge 0$;

(F₈) 

for all $x\in {\mathbb{R}}^3$, there exist $d,r>0$ such that $\tilde{F}(x,t):=f(x,t)t-2F(x,t)\ge d{\left|t\right|}^{\tau }$ for $\left|t\right|\ge r$.

Under the conditions $\left(F_6\right)-\left(F_8\right)$ and ${\left(SC\right)}^{\prime }$, the author proved two nontrivial solutions for system (32). The paper [29] studied more general superlinear terms (F₉) 

${lim}_{t\to \infty }{\displaystyle \frac{f(x,t)}{t}}=\infty .$

Influenced by the research methods, the potential V in [29] also needs the following condition:(V) 

$⟨\nabla V\left(x\right),x⟩\in L^2\left({\mathbb{R}}^3\right)$, $⟨\nabla V\left(x\right),x⟩\le 0$.

The authors in [29] proved the existence of two solutions for system (32) under the conditions ${\left(SC\right)}^{\prime }$ and $\left(F_9\right)$. Zhang in [30] replaces $\left(F_6\right)$ with the following local condition, i.e., (F₆)′ 

there exists $A\subset {\mathbb{R}}^3$, such that ${lim}_{\left|t\right|\to \infty }{\displaystyle \frac{F(x,t)}{t^2}}=+\infty$ for $x\in A$.

The author obtained a nontrivial solution for system (31) under the conditions ${\left(SC\right)}^{\prime },{\left(F_6\right)}^{\prime }$ and (F₁₀) 

for all $(x,t)\in {\mathbb{R}}^3\times \mathbb{R}$, $F(x,t)\ge 0$; there exists $C_1>0$ such that

$\mathcal{F}(x,t):={\displaystyle \frac{1}{2}}f(x,t)t-F(x,t)\ge \left({\displaystyle \frac{{\omega }^2}{8}}+C_1\right)t^2\ge 0,$

and there exist $C_2,R>0$ and $\varrho \in (0,1)$ such that

${\left[{\displaystyle \frac{\left|f\right(x,t\left)\right|}{{\left|t\right|}^{\varrho }}}\right]}^{{\displaystyle \frac{6}{5-\varrho }}}\le C_2\mathcal{F}(x,t),\left|t\right|\ge R.$

Li and Tang in [25] investigated the case where the nonlinear term exhibits sublinear growth and proved infinitely many solutions. Tang, Wen and Chen in [20] considered the case of critical growth and established the existence of nontrivial solutions.

Finally, the existence of sign-changing solutions for the Klein–Gordon–Maxwell system under coercive potentials is discussed. Zhang in [31] first studied the existence of sign-changing solutions for the Klein–Gordon–Maxwell system by the method of descending flow invariant sets. The main result are stated as follows.

Theorem 4.

Assume that $\left(SC\right)$ and $\left(AR\right)$ hold. If $\mu \ge 4$ or $\mu \in (2,4)$ and $0<\omega <\sqrt{{\displaystyle \frac{(\mu -2)V_0}{4-\mu }}}$, system (31) has one sign-changing solution. Meanwhile, if f is also odd, then system (31) has infinitely many sign-changing solutions, where $V_0:={inf}_{x\in {\mathbb{R}}^3}V\left(x\right)>0$.

Proof. 

First, define a positive cone and negative cone in the workspace H

$P^{+}:=\left\{u\in H:u\ge 0\right\},P^{-}:=\left\{u\in H:u\le 0\right\}.$

Then for any $\epsilon >$, set

$P_{\epsilon }^{+}:=\left\{u\in H:\mathrm{dist}(u,P^{+})<\epsilon \right\},P_{\epsilon }^{-}:=\left\{u\in H:\mathrm{dist}(u,P^{-})<\epsilon \right\}.$

For any given $u\in H$, the equation

$-\Delta v+V\left(x\right)v-(2\omega +{\varphi }_u){\varphi }_{u}v=f\left(u\right)$

has a unique solution $v\in H$, which we denote by $v=Au$. Then, A is continuous and compact, and A maps bounded set into bounded set. If f is odd, then A is also odd. By using the operator A, a locally Lipschitz continuous operator B can be constructed, which possesses the main properties of A. Notice that

$B\left(P_{\epsilon }^{+}\right)\subset P_{\epsilon }^{+},B\left(P_{\epsilon }^{-}\right)\subset P_{\epsilon }^{+}.$

Then the descent flow for the functional I can be established by operator B. By applying the quantitative deformation lemma within the frameworks of the mountain pass lemma and symmetric mountain pass lemma, the existence of sign-changing solutions and infinitely many sign-changing solutions can be obtained in the space $H\setminus \left(P_{\epsilon }^{+}\cup P_{\epsilon }^{-}\right)$. □

Later, Li, Tang and Sun in [32] generalized the conditions of [31] to the following fractional Klein–Gordon–Maxwell system:

(33)$\left\{\begin{array}{cc}{(-\Delta )}^{s}u+V\left(x\right)u-\left(2\omega +\varphi \right)\varphi u=f(x,u),\hfill & x\in {\mathbb{R}}^3,\hfill \\ {(-\Delta )}^s\varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3.\hfill \end{array}\right.$

The authors obtained that system (33) has infinitely many sign-changing solutions provided that $\mu \ge 4$ or $\mu \in (2,4)$ and $0<\omega <{\displaystyle \frac{\sqrt{(\mu -2)V_0}}{4-\mu }}$.

4.2. The Existence of Solutions for the Klein–Gordon–Maxwell System with a Steep Potential Well

In 1995, Bartch and Wang in [33] studied a class of potential $V_{\lambda }\left(x\right)=\lambda a\left(x\right)+1$ analogous to coercive potentials, where a satisfies the following conditions:(a₁)

$a\in \mathcal{C}({\mathbb{R}}^3,\mathbb{R})$, $a\ge 0$ and $a^{-1}\left(0\right)$ has a nonempty interior and smooth boundary.

(a₂)

There exists $a_0>0$ such that

$\mathrm{meas}\left(\{x\in {\mathbb{R}}^3:a\left(x\right)\le a_0\}\right)<\infty .$

We call $V_{\lambda }\left(x\right)$ a steep potential well as $\lambda \to \infty$ with the bottom $a^{-1}\left(0\right)$. Different from the coercive potential, we notice that the work space H under the conditions ($a_1)$ and ($a_2)$ can be no longer be embedded into $L^p\left({\mathbb{R}}^3\right)$$(2\le p<2^{*}={\displaystyle \frac{2N}{N-2}})$ compactly.

First, we introduce the existence of solutions for system (31) with a steep potential well when the nonlinearity exhibits subcritical growth, that is, f satisfies ${\left(SC\right)}^{\prime }$. Wang and Chen in [34] studied the case where f is super-cubic, namely, f satisfies $\left(F_4\right)$ and (f₁) 

for all $(x,t)\in {\mathbb{R}}^3\times \mathbb{R}$, $\mathcal{F}(x,t):={\displaystyle \frac{1}{4}}f(x,t)t-F(x,t)\ge 0$;

(f₂) 

for all $x\in {\mathbb{R}}^3$, there exist $a_1,L_1,\tau \in ({\displaystyle \frac{3}{2}},2)$ such that ${\left|f(x,t)\right|}^{\tau }\le a_1\mathcal{F}(x,t){\left|u\right|}^{\tau }$ for $\left|t\right|\ge L_1$.

Under the nonlinear term with a perturbation $h\left(x\right)$, ref. [34] gained two nontrivial solutions when $\lambda$ is large enough. Liu, Chen and Tang in [35] studied the case where the nonlinearity is superlinear, i.e., f satisfies ${\left(AR\right)}^{\prime }$, and proved the existence of ground states when $\lambda$ is large enough, if one of the following conditions holds:(i) 

$4\le \mu <6$;

(ii) 

$2<\mu <4$ and $0<\omega <{\displaystyle \frac{\sqrt{8(\mu -2)}}{4-\mu }}$.

Afterwards, based on the conditions of nonlinearity f in [35], Liu and Tang in [36] hypothesized f to be an odd function and got arbitrary k $(k\in \mathbb{N})$ pairs solutions. Zhang et al. in [37] also investigated the existence of ground state solutions for the system. The authors obtained a better result by utilizing the Pohozaev identity; that is, system (31) has ground states when $\lambda$ is large enough, provided one of the following conditions holds:(i) 

$3\le \mu <6$;

(ii) 

$2<\mu <3$ and $0<\omega <{\displaystyle \frac{\sqrt{(\mu -2)(4-\mu )}}{3-\mu }}$.

Liu, Kang and Tang in [38] studied the case where the nonlinearity is asymptotically linear, i.e., f satisfies (f₃) 

for all $(x,t)\in {\mathbb{R}}^3\times \mathbb{R}$, there exists $\overline{C}>0$ such that $\left|f(x,t)\right|\le \overline{C}\left|t\right|$,

(f₄) 

for all $x\in {\mathbb{R}}^3$, there exists $k\in \{1,2,\dots \}$ such that

${\nu }_k<\underset{\left|t\right|\to 0}{lim\; inf}{\displaystyle \frac{f(x,t)}{t}}\le \underset{\left|t\right|\to 0}{lim\; sup}{\displaystyle \frac{f(x,t)}{t}}<{\nu }_{k+1},$

(f₅) 

${lim\; sup}_{\left|t\right|\to \infty }{\displaystyle \frac{f(x,t)}{t}}<{\nu }_1,$

Here ${\nu }_i$ is the eigenvalue of the following equation:

$\left\{\begin{array}{cc}-\Delta u+u=\nu u,\hfill & x\in \Omega ,\hfill \\ u=0,\hfill & x\in \partial \Omega .\hfill \end{array}\right.$

The paper [38] obtained the positive solution $u_{\lambda ,\omega }$ and studied the decay properties of $u_{\lambda ,\omega }$ at infinity; i.e., there exist $A,{\mathrm{\Lambda }}_0,R_0>0$ such that

$u_{\lambda ,\omega }\le A{\lambda }^{-{\displaystyle \frac{1}{2}}}{e}^{-{\left({\displaystyle \frac{a_0}{2}}\lambda \right)}^{{\displaystyle \frac{1}{2}}}(|x|-R_0)}\mathrm{for}\left|x\right|>R_0.$

Moreover, [38] investigated the asymptotic behavior of the positive solution $u_{\lambda ,\omega }$, namely

$u_{\lambda ,\omega }\to u_{\lambda }$ as $\omega \to 0$, where $u_{\lambda }$ is a positive solution for the following equation:

$-\Delta u+(\lambda a\left(x\right)+1)u=f(x,u),x\in {\mathbb{R}}^3;$

$u_{\lambda ,\omega }\to u_{\omega }$ as $\lambda \to \infty$, where $u_{\omega }$ is a positive solution for the following system:

$\left\{\begin{array}{cc}-\Delta u+u-(2\omega +\varphi )\varphi u=f(x,u),\hfill & x\in \Omega ,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in \Omega ,\hfill \end{array}\right.$

and $u_{\omega }\left(x\right)=0$ a.e. for $x\in {\mathbb{R}}^3\setminus \overline{\Omega }$;

$u_{\lambda ,\omega }\to u_{*}$ as $\lambda \to \infty ,\omega \to 0$, where $u_{*}$ is a positive solution for the following equation:

$-\Delta u+u=f(x,u),x\in \Omega .$

When the nonlinearity f exhibits critical growth, Zhang in [39] considered the following Klein–Gordon–Maxwell system:

(34)$\left\{\begin{array}{cc}-\Delta u+V_{\lambda }\left(x\right)u-\left(2\omega +\varphi \right)\varphi u=\theta g\left(u\right)+u^5,\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3,\hfill \end{array}\right.$

where g satisfies (G₁) 

${lim}_{t\to 0^{+}}{\displaystyle \frac{g\left(t\right)}{t}}={lim}_{t\to \infty }{\displaystyle \frac{g\left(t\right)}{t^5}}=0$;

(G₂) 

there exist $\nu \in (4,6),D>0,\rho >0$ such that $G\left(t\right)\ge {\displaystyle \frac{D}{{\rho }^{\nu }}}{t}^{\nu }$;

(G₃) 

$g\left(t\right)t-4G\left(t\right)\ge 0$.

The paper [39] obtained a nontrivial solution provided that $\lambda ,\theta$ are sufficiently large. Combined with the Pohozaev identity, Gan and Wang in [40] further investigated the existence of solutions for system (34) and obtained the following theorem.

Theorem 5.

Assume that $\left(a_1\right)-\left(a_3\right)$, $\left(G_1\right)-\left(G_2\right)$, $\left(AR\right)$ and $G\left(t\right)\ge 0$ for $t\ge 0$, where(a₃) 

for all $x\in {\mathbb{R}}^3$, $⟨\nabla A\left(x\right),x⟩\ge 0$.

Then system (34) has a ground state solution, if one of the following conditions holds: (i) 

$4<\mu <6$, $\theta >0$;

(ii) 

$3\le \mu \le 4$, θ sufficiently large;

(iii) 

$2<\mu <3$, $0<\omega <{\displaystyle \frac{\sqrt{(4-\mu )(\mu -2)}}{3-\mu }}$, θ sufficiently large.

Proof. 

$Step1.$ Construct a sequence of approximate solutions, that is, a $\left(PPS\right)$ sequence $\left\{u_n\right\}$ satisfying

(35)$\begin{array}{c}\hfill I_{\lambda }\left(u_n\right)\to c_{\lambda },I_{\lambda }^{\prime }\left(u_n\right)\to 0,\mathcal{P}\left(u_n\right)\to 0,n\to \infty ,\end{array}$

where $\mathcal{P}\left(u\right)=0$ is the Pohozaev identity corresponding to system (34) and $c_{\lambda }$ is the mountain pass energy level for the functional $c_{\lambda }$.

$Step2.$ When $4<\mu <6$ and $\theta >0$ or $2<\mu \le 4$ and $\theta$ sufficiently large, one obtains

(36)$\begin{array}{c}\hfill 0<c_{\lambda }<{\displaystyle \frac{1}{3}}{S}^{{\displaystyle \frac{3}{2}}}.\end{array}$

$Step3.$ If $\left\{u_n\right\}$ satisfies (35), then $\left\{u_n\right\}$ is bounded. From (36), one sees

(37)$\begin{array}{c}\hfill \underset{n\to \infty }{lim}\underset{y_n\in {\mathbb{R}}^3}{sup}{\int }_{{\mathbb{R}}^3}{u}_n^{2}dx=\kappa >0.\end{array}$

Let

$A_R:=\{x\in {\mathbb{R}}^3:\left|x\right|>R,a\left(x\right)\ge a_0\},B_R:=\{x\in {\mathbb{R}}^3:\left|x\right|>R,a\left(x\right)<a_0\}.$

Since $\left\{u_n\right\}$ is bounded, for $\lambda >{\displaystyle \frac{4C}{\kappa a_0}}$, we get

$\begin{array}{c}\hfill {\int }_{A_R}{u}_n^{2}dx\le {\displaystyle \frac{1}{\lambda a_0+1}}{\int }_{A_R}(\lambda a\left(x\right)+1)u_n^{2}dx\le {\displaystyle \frac{C}{\lambda a_0}}\le {\displaystyle \frac{\kappa }{4}}.\end{array}$

Since $\mathrm{meas}\left(B_R\right)\to 0$ as $R\to \infty$ by $\left(a_2\right)$, then

$\begin{array}{c}\hfill {\int }_{B_R}{u}_n^{2}dx\le {\left({\int }_{{\mathbb{R}}^3}{u}_n^{6}dx\right)}^{{\displaystyle \frac{1}{3}}}{\left({\int }_{B_R}1dx\right)}^{{\displaystyle \frac{2}{3}}}\le C{\left(\mathrm{meas}\left(B_R\right)\right)}^{{\displaystyle \frac{2}{3}}}\le {\displaystyle \frac{\kappa }{4}}.\end{array}$

If $u_n\to 0$ in $L_{loc}^p\left({\mathbb{R}}^3\right)$, then

$\begin{array}{c}\hfill \kappa =\underset{n\to \infty }{lim}\underset{y_n\in {\mathbb{R}}^3}{sup}{\int }_{{\mathbb{R}}^3}{u}_n^{2}dx\le \underset{n\to \infty }{lim}{\int }_{{\mathbb{R}}^3}{u}_n^{2}dx=\underset{n\to \infty }{lim}\left({\int }_{B_R}{u}_n^{2}dx+{\int }_{B_R^c}{u}_n^{2}dx\right)\le {\displaystyle \frac{\kappa }{2}},\end{array}$

which derives a contradiction. Hence, there exists $u\in H\setminus \left\{0\right\}$ such that $I_{\lambda }^{\prime }\left(u\right)=0$; namely, the set $\mathcal{M}:=\{u\in H\setminus \left\{0\right\}:I_{\lambda }^{\prime }\left(u\right)=0\}$ is nonempty.

$Step4.$ Define $m:={inf}_{u\in \mathcal{M}}{I}_{\lambda }\left(u\right)$. Let $\left\{u_n\right\}\subset \mathcal{M}$ be a minimizing sequence; then $\left\{u_n\right\}$ satisfies (35), which implies $\left\{u_n\right\}$ is bounded. Thus, there exists $u_0$ such that $I_{\lambda }^{\prime }\left(u_0\right)=0$ and $I_{\lambda }\left(u_0\right)\ge m$. Together with Fatou’s lemma, we get that $I_{\lambda }\left(u_0\right)=m$. □

Replace $\theta$ in system (34) with $K\left(x\right)$, where $K\left(x\right)$ satisfies the following conditions:(K₁) 

$K\in \mathcal{C}({\mathbb{R}}^3,\mathbb{R})$, $0<K_0:={inf}_{x\in {\mathbb{R}}^3}K\left(x\right)\le K\left(x\right)\le K_{\infty }:={sup}_{x\in {\mathbb{R}}^3}K\left(x\right)<\infty$;

(K₂) 

for all $x\in {\mathbb{R}}^3$, $⟨\nabla K\left(x\right),x⟩\le 0$.

Similar to Theorem 5, ref. [40] also gained the existence of the ground state solution.

Particularly, Tang, Wen and Chen in [20] considered the semiclassical problem

(38)$\left\{\begin{array}{cc}-{\epsilon }^2\Delta u+V\left(x\right)u-\left(2\omega +\varphi \right)\varphi u=g(x,u)+K\left(x\right)u^5,\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3.\hfill \end{array}\right.$

By using Lemma 2, system (38) is reduced to a single equation

(39)$-{\epsilon }^2\Delta u+V\left(x\right)u-\left(2\omega +{\varphi }_u\right){\varphi }_{u}u=g(x,u)+K\left(x\right)u^5.$

Let $\lambda ={\epsilon }^{-2}$; then (39) becomes the following equation:

(40)$-\Delta u+\lambda V\left(x\right)u-\lambda \left(2\omega +{\varphi }_u\right){\varphi }_{u}u=\lambda g(x,u)+\lambda K\left(x\right)u^5.$

Since $V\left(x\right)$ satisfies $\left(a_1\right)-\left(a_2\right)$ and $\epsilon \to 0$, the semiclassical problem can be transformed into a problem with a steep potential well for resolution. The main result for the system (38) in [20] is as follows.

Theorem 6.

Assume that $V\left(0\right)=0$, $\left(a_1\right)-\left(a_2\right),\left(K_1\right),{\left(SC\right)}^{\prime }$, $\left(AR\right)$ and $\left(G_4\right)$ hold, where (G₄) 

there exist $\alpha ,r,T>0$, $p\in (2,4]$ such that $G(x,t)\ge \alpha t^p$ for $\left|x\right|\le r$, $0\le t\le T$.

Then there exists ${\epsilon }_0>0$ such that system (38) admits a nontrivial solution for $\epsilon \in (0,{\epsilon }_0]$, provided one of the following conditions holds: (i) 

$4\le \mu <6$;

(ii) 

$2<\mu <4$, $0<\omega <{\displaystyle \frac{2\sqrt{2(\mu -2)a_0}}{4-\mu }}$.

4.3. The Existence of Solutions for the Klein–Gordon–Maxwell System with Periodic Potential

We call V the periodic potential if the following conditions hold:(W₁) 

$V(x+y)=V\left(x\right),x\in {\mathbb{R}}^3,y\in {\mathbb{Z}}^3$;

(W₂) 

$V\left(x\right)\ge V_0>0$.

First, we introduce the existence of solutions for system (31) with periodic potential when the nonlinearity is subcritical. Cunha in [41] considered the case where f is super-cubic; namely, f satisfies $\left(SC\right)$, $f\left(0\right)=0$ and (F₄)′ 

${lim}_{\left|t\right|\to \infty }{\displaystyle \frac{F\left(t\right)}{t^4}}=+\infty$;

(P₁) 

${\displaystyle \frac{f\left(t\right)}{t^3}}$ is increasing for $\left|t\right|>0$,

They obtained the existence of a ground state solution. By replacing condition $\left(P_1\right)$ with the weaker condition $\left(f_1\right)$, ref. [41] also gained a ground state solution. By the property $-\omega \le {\varphi }_u\le 0$ and a more refined calculations, Chen and Tang in [11] studied the case where the nonlinearity f is superlinear; i.e., f satisfies ${\left(SC\right)}^{\prime }$, ${\left(AR\right)}^{\prime }$, $\left(F_6\right)$ and

(P₂) 

for all $x\in {\mathbb{R}}^3$, there exists $r>0$ such that $F(x,t)\ge 0$ for $\left|t\right|\ge r$.

The authors proved the existence of ground state solutions for system (31), if one of the following conditions holds:(i) 

$4\le \mu <6$;

(ii) 

$2<\mu <4$, ${\displaystyle \frac{V_0}{{\omega }^2}}\ge {\displaystyle \frac{{(4-\mu )}^2}{8(\mu -2)}}$.

Later, based on the conditions of nonlinearity f in [11], Chen and Tang in [42] hypothesized f to be an odd function and obtained infinitely many pairs $\pm u$ of geometrically distinct solutions. Zhang et al. in [30] investigated a weaker periodic potential by replacing $\left(W_2\right)$ with ${\left(W_2\right)}^{\prime }$:(W₂)′ 

$sup\left[\sigma (-\Delta +V)\cap (-\infty ,0)\right]<0<\mathrm{\Theta }:=inf\left[\sigma (-\Delta +V)\cap (0,+\infty )\right]$.

The main result in [30] can be stated.

Theorem 7.

When f satisfies ${\left(SC\right)}^{\prime }$, ${\left(F_6\right)}^{\prime }$ and(P₃) 

$F(x,t)\ge 0$ for all $(x,t)\in {\mathbb{R}}^3\times \mathbb{R}$; there exists $C_1>0$ such that

$\mathcal{F}(x,t):={\displaystyle \frac{1}{2}}f(x,t)t-F(x,t)\ge \left({\displaystyle \frac{{\omega }^2}{8}}+C_1\right)t^2\ge 0,$

and there exist $C_2,{\delta }_0\in (0,\mathrm{\Theta })$ and $\varrho \in (0,1)$ such that

${\displaystyle \frac{f(x,t)}{t}}\ge \mathrm{\Theta }-{\delta }_0\mathrm{implies}{\left[{\displaystyle \frac{\left|f\right(x,t\left)\right|}{{\left|t\right|}^{\varrho }}}\right]}^{{\displaystyle \frac{6}{5-\varrho }}}\le C_2\mathcal{F}(x,t),$

system (31) has a nontrivial solution.

Proof. 

By ${\left(W_2\right)}^{\prime }$, the potential V is sign-changing and then the workspace has the orthogonal decomposition $H=H^{+}\oplus H^{-}$. Define the energy functional

$\begin{array}{c}\hfill I\left(u\right)={\displaystyle \frac{1}{2}}\left(\parallel u^{+}{\parallel }^2-{\parallel u^{-}\parallel }^2\right)-\mathrm{\Psi }\left(u\right),u=u^{+}+u^{-}\in H^{+}\oplus H^{-},\end{array}$

where

$\mathrm{\Psi }\left(u\right):=\left({\int }_{{\mathbb{R}}^3}{\displaystyle \frac{1}{2}}\omega {\varphi }_u{u}^{2}dx+F(x,u)dx\right).$

From ${\left(SC\right)}^{\prime }$, ${\left(F_6\right)}^{\prime }$ and $\left(P_3\right)$, $\mathrm{\Psi }\in {\mathcal{C}}^1$ is bounded from below and weakly sequentially lower semi-continuous and ${\mathrm{\Psi }}^{\prime }$ is weakly sequentially continuous. By ${\left(SC\right)}^{\prime }$, there exists $\rho >0$ such that

$k:=inf\left\{I\left(u\right):u\in H^{+},\parallel u\parallel =\rho \right\}>0.$

From ${\left(F_6\right)}^{\prime }$, one can assume that A is a bounded domain without loss of generality. Choose $w\in {\mathcal{C}}_0^{\infty }(A,{\mathbb{R}}^{+})\cap {\mathcal{C}}_0^{\infty }({\mathbb{R}}^3,{\mathbb{R}}^{+})$ such that

$\parallel u^{+}{\parallel }^2-{\parallel u^{-}\parallel }^2={\int }_{{\mathbb{R}}^3}\left({|\nabla w|}^2+V\left(x\right)w^2\right)dx={\int }_A\left({|\nabla w|}^2+V\left(x\right)w^2\right)dx\ge 1.$

It follows from ${\left(SC\right)}^{\prime }$ and ${\left(F_6\right)}^{\prime }$ that $supI(H^{-}\oplus {\mathbb{R}}^{+}{w}^{+})<\infty .$ Then there exists $R_w>0$ such that

$I\left(u\right)\le 0,u\in H^{-}\oplus {\mathbb{R}}^{+}{w}^{+},\parallel u\parallel \ge R_w.$

Hence, there exists $r>\rho$ such that $sup(\partial Q)\le 0$, where

$Q=\left\{v+s{w}^{+}:v\in H^{-},s\ge 0,\parallel v+s{w}^{+}\parallel \ge r\right\}.$

From the generalized linking theorem in [43,44], there exist $c\in [k,supI(Q\left)\right]$ and a sequence $\left\{u_n\right\}$ such that

$I\left(u_n\right)\to c,I^{\prime }\left(u_n\right)\to 0,n\to \infty .$

If

$\delta :=\underset{n\to \infty }{lim\; sup}{\int }_{B_1\left(y\right)}{u}_n^{2}dx=0,$

the $u_n\to 0$ in $L^s\left({\mathbb{R}}^3\right)(2<s<2^{*})$. By ${\left(SC\right)}^{\prime }$, one sees

${\int }_{{\mathbb{R}}^3}\left[{\displaystyle \frac{1}{2}}f\left(u_n\right)u_n-F\left(u_n\right)\right]dx=o\left(1\right),$

Thus,

$\begin{array}{cc}\hfill c+o\left(1\right)& =I\left(u_n\right)-{\displaystyle \frac{1}{2}}⟨I^{\prime }\left(u_n\right),u_n⟩\hfill \\ \hfill & =-{\displaystyle \frac{1}{2}}{\parallel \nabla {\varphi }_{u_n}\parallel }_{L^2\left({\mathbb{R}}^3\right)}^2+{\int }_{{\mathbb{R}}^3}\left[{\displaystyle \frac{1}{2}}f\left(u_n\right)u_n-F\left(u_n\right)\right]dx\hfill \\ \hfill & \le {\int }_{{\mathbb{R}}^3}\left[{\displaystyle \frac{1}{2}}f\left(u_n\right)u_n-F\left(u_n\right)\right]dx=o\left(1\right),\hfill \end{array}$

which is a contradiction. Then there exists $y_n\in {\mathbb{R}}^3$ such that ${\int }_{B_1\left(y_n\right)}{u}_n^{2}dx>{\displaystyle \frac{\delta }{2}}$. Set $v_n\left(x\right)=u_n(x+y_n)$. The boundedness of $\left\{v_n\right\}$ can be gained by ${\left(SC\right)}^{\prime }$, ${\left(F_6\right)}^{\prime }$ and $\left(P_3\right)$. Then there exists $v\in H$ such that $v_n\rightharpoonup v\ne 0$. □

When the nonlinearity f exhibits critical growth, Carrião in [2] studied the following Klein–Gordon–Maxwell system:

(41)$\left\{\begin{array}{cc}-\Delta u+V\left(x\right)u-\left(2\omega +\varphi \right)\varphi u={\theta \left|u\right|}^{q-2}u+{\left|u\right|}^{2^{*}-2}u,\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3.\hfill \end{array}\right.$

They proved the existence of positive ground states if one of the following conditions holds:(i) 

$4<q<6$, $\theta >0$;

(ii) 

$q=4$, $\theta$ sufficiently large;

(iii) 

$2<q<4$, ${\displaystyle \frac{V_0}{{\omega }^2}}\ge {\displaystyle \frac{2(4-q)}{q-2}}$, $\theta$ sufficiently large.

Wang in [45] further investigated the case $2<q<4$ in [2]. The authors obtained that system (41) has a ground state solution provided that $2<q<4$, ${\displaystyle \frac{V_0}{{\omega }^2}}\ge {\displaystyle \frac{{(4-q)}^2}{2(q-2)}}$ and $\theta$ is sufficiently large. Later, Tang, Wen and Chen in [20] further improved the conditions in [45]. The authors gained the same result when $2<q<4$, ${\displaystyle \frac{V_0}{{\omega }^2}}\ge {\displaystyle \frac{{(4-q)}^2}{8(q-2)}}$ and $\theta$ is sufficiently large. Evidently,

${\displaystyle \frac{{(4-q)}^2}{8(q-2)}}\le {\displaystyle \frac{{(4-q)}^2}{2(q-2)}}\le {\displaystyle \frac{2(4-q)}{q-2}}.$

Moreover, Tang, Wen and Chen in [20] generalized the nonlinear term ${\left|u\right|}^{p-2}u$ in system (41) to the general term $g\left(u\right)$, namely

(42)$\left\{\begin{array}{cc}-\Delta u+V\left(x\right)u-\left(2\omega +\varphi \right)\varphi u=\theta g\left(u\right)+{\left|u\right|}^{2^{*}-2}u,\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3,\hfill \end{array}\right.$

where g satisfies $\left(SC\right)$, $\left(AR\right)$ and (G₄)′ 

if $\mu \in (2,4]$, there exist $\alpha >0$ and $p\in (2,4]$ such that $G\left(t\right)\ge \alpha t^p$ for $t\ge 1$.

The authors proved the existence of ground states for system (42) if one of the following conditions holds:(i) 

$4<\mu <6$, $\theta >0$;

(ii) 

$\mu =4$, $\theta$ sufficiently large;

(iii) 

$2<\mu <4$, ${\displaystyle \frac{V_0}{{\omega }^2}}\ge {\displaystyle \frac{{(4-q)}^2}{8(q-2)}}$, $\theta$ sufficiently large.

4.4. The Existence of Solutions for the Klein–Gordon–Maxwell System with Vanishing Potential

Chen and Li in [46] considered the following Klein–Gordon–Maxwell with vanishing potential:

(43)$\left\{\begin{array}{cc}-\Delta u+V\left(x\right)u-\left(2\omega +\varphi \right)\varphi u=K\left(x\right){\left|u\right|}^{p-2}u,\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3,\hfill \end{array}\right.$

where $V,K\in \mathcal{C}({\mathbb{R}}^3,\mathbb{R})$ are radial functions and satisfy (VK₁) 

there exist $a,b,\beta ,A>0$ and $\alpha \in (0,2]$ such that

${\displaystyle \frac{a}{1+{\left|x\right|}^{\alpha }}}\le V\left(x\right)\le A;0<K\left(x\right)<{\displaystyle \frac{b}{1+{\left|x\right|}^{\beta }}}.$

The paper [46] gained the existence of infinitely many radial solutions when $\alpha \in (0,{\displaystyle \frac{4}{11}})$, $p\in (4,6)\cap (\sigma ,6)$, where if $0<\beta <\alpha$, $\sigma =6-{\displaystyle \frac{4\beta }{\alpha }}$; otherwise, $\sigma =2$. Elson et al. in [47] studied the general case f, namely

(44)$\left\{\begin{array}{cc}-\Delta u+V\left(x\right)u-\left(2\omega +\varphi \right)\varphi u=K\left(x\right)f\left(u\right),\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3,\hfill \end{array}\right.$

where $V,K\in \mathcal{C}({\mathbb{R}}^3,\mathbb{R})$ and $K\in L^{\infty }\left({\mathbb{R}}^3\right)$ satisfy (VK₂) 

for all $x\in {\mathbb{R}}^3$, there exist $a_1,a_2,{\xi }_0,V_0$ such that

$0<V_0\le V\left(x\right)\le a_1,$

If $2<\theta <4$, then

$V_0\ge {\displaystyle \frac{2(4-\theta )}{\theta -2}}>0;$

and

$0<K\left(x\right)<{\displaystyle \frac{a_2}{1+{\left|x\right|}^{{\xi }_0}}},$

(VK₃) 

if $\left\{{\mathcal{A}}_n\right\}\subset {\mathbb{R}}^3$ is a sequence of Borel sets such that there exists $R>0$ with $\mathrm{meas}\left({\mathcal{A}}_n\right)\le R$, then

$\underset{r\to \infty }{lim}{\int }_{{\mathcal{A}}_n\cap B_r^c\left(0\right)}K\left(x\right)dx=0.$

f is super-cubic; i.e., f satisfies $\left(SC\right)$ and (F₁)′ 

for all $t\in \mathbb{R}$, there exists $\mu >4$ such that $\mu F\left(t\right)\ge tf\left(t\right).$

The authors obtained the existence of the positive ground state solution for system (44). Afterwards, Gan et al. in [48] utilized the knowledge related to the Pohozaev identity to study the case where the nonlinearity f in system (44) is superlinear. Assume that f satisfies $\left(SC\right)-\left(AR\right)$ and $V,K\in \mathcal{C}({\mathbb{R}}^3,\mathbb{R})$, $K\in L^{\infty }\left({\mathbb{R}}^3\right)$ satisfy $\left(V{K}_3\right)$ and

(VK₄) 

for all $x\in {\mathbb{R}}^3$; there exist $\kappa ,a_1,a_2,a_3,{\xi }_0$ such that

${\displaystyle \frac{a_1}{1+{\left|x\right|}^{\kappa }}}\le V\left(x\right)\le a_2;0<K\left(x\right)<{\displaystyle \frac{a_3}{1+{\left|x\right|}^{{\xi }_0}}},$

(VK₅) 

$⟨\nabla V\left(x\right),x⟩\in L^2\left({\mathbb{R}}^3\right)$, $⟨\nabla V\left(x\right),x⟩\ge 0$; $⟨\nabla K\left(x\right),x⟩\in L^{\infty }\left({\mathbb{R}}^3\right)$, $⟨\nabla K\left(x\right),x⟩\le 0$.

The main result in [48] is as follows.

Theorem 8.

System (44) has a positive ground state solution, provided that one of the following conditions holds: (i) 

$3\le \mu <6$;

(ii) 

$2<\mu <3$, $0<\omega <{\displaystyle \frac{\sqrt{(\mu -2)(4-\mu )}}{3-\mu }}$.

Proof. 

Define the energy functional corresponding to system (44)

$\begin{array}{c}\hfill I\left(u\right)={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^3}\left({|\nabla u|}^2+V\left(x\right)u^2\right)dx-{\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^3}\omega {\varphi }_u{u}^{2}dx-{\int }_{{\mathbb{R}}^3}K\left(x\right)F\left(u\right)dx.\end{array}$

Clearly, I satisfies the mountain pass structure. Similar to the proof of (23) in Theorem 1, a (PPS) sequence $\left\{u_n\right\}$ corresponding to I can be obtained as follows:

(45)$\begin{array}{c}\hfill I\left(u_n\right)\to c,I^{\prime }\left(u_n\right)\to 0,\mathcal{P}\left(u_n\right)\to 0,n\to \infty ,\end{array}$

where $\mathcal{P}\left(u\right)=0$ is the Pohozaev identity corresponding to system (44) and c is the mountain pass energy level for the functional I. Combining the property $-\omega \le {\varphi }_u\le 0$ and $\left(i\right)-\left(ii\right)$, the sequence $\left\{u_n\right\}$ is bounded. For all $s\in (2,6)$, the embedding from workspace H to $L_K^s\left({\mathbb{R}}^3\right)$ is compact, where

$L_K^s\left({\mathbb{R}}^3\right):=\left\{u:{\mathbb{R}}^3\to \mathbb{R}|uisameasurablefunctionand{\int }_{{\mathbb{R}}^3}K\left(x\right)F\left(u\right)dx<\infty \right\}.$

Define the ground state energy and solution set

$m:=\underset{u\in \mathcal{M}}{inf}I\left(u\right),\mathcal{M}:=\{u\in H^1\left({\mathbb{R}}^3\right)\setminus \left\{0\right\}:I^{\prime }\left(u\right)=0\}.$

Similar to the proof of Theorem 2, there exists $0\ne \tilde{u}\in \mathcal{M}$ such that $I\left(\tilde{u}\right)=m$. □

4.5. The Existence of Solutions for the Klein–Gordon–Maxwell System with Radial Potential

Xu and Chen in [49] considered the following Klein–Gordon–Maxwell system:

(46)$\left\{\begin{array}{cc}-\Delta u+u-\left(2\omega +\varphi \right)\varphi u={\left|u\right|}^{p-2}u+h\left(x\right),\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3,\hfill \end{array}\right.$

where h satisfies (h₁) 

$h\in {\mathcal{C}}^1\left({\mathbb{R}}^3\right)\cap L^2\left({\mathbb{R}}^3\right)$$h\left(x\right)=h\left(\right|x\left|\right)\neg \equiv 0$ and $⟨\nabla h\left(x\right),x⟩\in L^2\left({\mathbb{R}}^3\right)$;

(h₂) 

${\parallel h\left(x\right)\parallel }_{L^2\left({\mathbb{R}}^3\right)}<{\displaystyle \frac{p-1}{2p}}{\left({\displaystyle \frac{p+1}{2p{S}^{p+1}}}\right)}^{{\displaystyle \frac{1}{p-1}}}$.

They proved the existence of two nontrivial solutions for system (46) if $p\in (3,6)$ or $p\in (2,3]$ and $\omega$ is small enough. Liu and Wu in [50] investigated the following Klein–Gordon–Maxwell system:

(47)$\left\{\begin{array}{c}-\Delta u-(2\omega +\varphi )\varphi u=g\left(u\right)+h\left(x\right),\mathrm{in}{\mathbb{R}}^3,\hfill \\ \Delta \varphi =(\omega +\varphi )u^2,\mathrm{in}{\mathbb{R}}^3,\hfill \end{array}\right.$

where g satisfies (BL) conditions $\left(g_1\right)-\left(g_3\right)$ and h satisfies (h₁) 

$(x·\nabla h)\in L^{{\displaystyle \frac{6}{5}}}\left({\mathbb{R}}^3\right)$, where $\nabla h$ is in weak sense;

(h₂) 

$h\in L^2\left({\mathbb{R}}^3\right)$ is a radial function and $h\left(x\right)\neg \equiv 0$.

Then, there exist ${\omega }_0>0$ such that system (47) has two nontrivial solutions for $0<\omega <{\omega }_0$. Sun, Duan and Liu in [51] studied the following Klein–Gordon–Maxwell system:

(48)$\left\{\begin{array}{cc}-\Delta u+u-\left(2\omega +\varphi \right)\varphi u=f(x,u),\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^3,\hfill \end{array}\right.$

where f satisfies $\left(SC\right)$, $\left(F_6\right)$ and the following conditions:(Q) 

for all $(x,t)\in {\mathbb{R}}^3\times \mathbb{R}$, there exist $C>0$ such that $f(x,t)t-2F(x,t)\ge -{C\left|t\right|}^2$.

The authors gained the existence of a nontrivial solution for system (48). Further, if f is odd, then the existence of infinitely many solutions system can be obtained.

5. The Existence of Solutions for the Klein–Gordon–Maxwell System in ${\mathbb{R}}^{\mathbf{2}}$

In this chapter, we consider the following Klein–Gordon–Maxwell system in ${\mathbb{R}}^2$:

(49)$\left\{\begin{array}{cc}-\Delta u+V\left(x\right)u-\left(2\omega +\varphi \right)\varphi u=f\left(u\right),\hfill & x\in {\mathbb{R}}^2,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^2.\hfill \end{array}\right.$

The energy functional corresponding to system (49) is

(50)$\begin{array}{c}\hfill F(u,\varphi )={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^2}\left\{{|\nabla u|}^2+V\left(x\right)u^2-{|\nabla \varphi |}^2-(2\omega +\varphi )\varphi u^2\right\}dx-{\int }_{{\mathbb{R}}^2}F\left(u\right)dx.\end{array}$

Following a similar approach as in the nonlocal term ${\varphi }_u$ of the Klein–Gordon–Maxwell system on ${\mathbb{R}}^3$, one knows that there exists a unique $\varphi ={\varphi }_u$, which solves the second equation of system (49) for any fixed $u\in H^1\left({\mathbb{R}}^2\right)$, and $-\omega \le {\varphi }_u\le 0$. Thus, the functional (50) can be transformed into

$\begin{array}{c}\hfill I\left(u\right)={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^2}\left({|\nabla u|}^2+V\left(x\right)u^2\right)dx-{\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^2}\omega {\varphi }_u{u}^{2}dx-{\int }_{{\mathbb{R}}^2}F\left(u\right)dx.\end{array}$

Albuquerque and Li in [52] first studied the following Klein–Gordon–Maxwell system in ${\mathbb{R}}^2$:

(51)$\left\{\begin{array}{cc}-\Delta u+V\left(x\right)u-K\left(x\right)\left(2\omega +\varphi \right)\varphi u=K\left(x\right)f\left(u\right),\hfill & x\in {\mathbb{R}}^2,\hfill \\ -\Delta \varphi =-K\left(x\right)(\omega +\varphi )u^2,\hfill & x\in {\mathbb{R}}^2,\hfill \end{array}\right.$

where $V,K:(0,\infty )\to \mathbb{R}$ is positive, radial, continuous and satisfies (U₁) 

there exist $a_0,a_{\infty }>-2$ such that

$\underset{r\to +\infty }{lim\; inf}{\displaystyle \frac{V\left(r\right)}{r^{a_{\infty }}}}>0,\underset{r\to 0^{+}}{lim\; inf}{\displaystyle \frac{V\left(r\right)}{r^{a_0}}}>0,$

(U₂) 

there exist $b_0,b_{\infty }$ and $b_{\infty }<a_{\infty },b_0>-2$ such that

$\underset{r\to +\infty }{lim\; sup}{\displaystyle \frac{K\left(r\right)}{r^{b_{\infty }}}}<\infty ,\underset{r\to 0^{+}}{lim\; sup}{\displaystyle \frac{K\left(r\right)}{r^{b_0}}}<\infty .$

The paper [52] investigated the case where the nonlinearity has critical growth, i.e., f satisfies ${\left(F_1\right)}^{\prime }$ and (R₁) 

for all $s\in \mathbb{R}$, there exist $a_0,b_1,b_2>0$ such that $\left|f\left(s\right)\right|\le b_1\left|s\right|+b_2\left(e^{{\alpha }_0{s}^2}-1\right)$;

(R₂) 

for all $s\in \mathbb{R}$, there exist ${\mu }_0>2,\theta >0$ such that $F\left(s\right)\ge {\displaystyle \frac{\theta }{{\mu }_0}}{\left|s\right|}^{{\mu }_0}$;

(R₃) 

${lim}_{s\to 0^{+}}{\displaystyle \frac{2F\left(s\right)}{s^2}}=0$.

The authors proved that there exists ${\theta }_0>0$ such that system (51) has a nontrivial solution when $\theta >{\theta }_0$. Chen, Lin and Tang in [53] investigated the existence and nonexistence of solutions for system (49) under three cases of potential V: radial potential, coercive potential and a constant. Firstly, ref. [53] studied the existence of solutions for system (49) under the critical growth, i.e., $f\in \mathcal{C}(\mathbb{R},\mathbb{R})$ satisfies $\left(AR\right)$ and (R₄) 

there exists ${\alpha }_0>0$ such that ${lim}_{\left|t\right|\to \infty }{\displaystyle \frac{\left|f\right(t\left)\right|}{e^{\alpha t^2}}}=0$ for $\alpha >{\alpha }_0$; ${lim}_{\left|t\right|\to \infty }{\displaystyle \frac{\left|f\right(t\left)\right|}{e^{\alpha t^2}}}=+\infty$ for $\alpha <{\alpha }_0$.

(R₅) 

${lim}_{t\to 0}{\displaystyle \frac{f\left(t\right)}{t}}=0$.

(R₆) 

there exist $M_0,t_0>0$ such that $F\left(t\right)\le M_0\left|f\left(t\right)\right|$ for $\left|t\right|\ge t_0$.

(R₇) 

${lim\; inf}_{\left|t\right|\to \infty }{\displaystyle \frac{t^{2}F\left(t\right)}{e^{{\alpha }_0{t}^2}}}\ge \kappa >{\displaystyle \frac{4}{{\alpha }_0^2{\rho }^2}}$, where $\rho >0$ and ${\rho }^2\left[{max}_{\left|x\right|\le \rho }V\left(x\right)+{\omega }^2\right]<2$.

Assume that f satisfies $\left(R_4\right)-\left(R_7\right)$ and $\left(AR\right)$; V satisfies (U₃) 

$V\in \mathcal{C}({\mathbb{R}}^2,\mathbb{R}),V\left(x\right)=V\left(\right|x\left|\right)$, ${inf}_{u\in H^1\left({\mathbb{R}}^2\right)\setminus \left\{0\right\}}{\displaystyle \frac{{\int }_{{\mathbb{R}}^2}\left({|\nabla u|}^2+V\left(x\right)u^2\right)dx}{{\int }_{{\mathbb{R}}^2}{u}^{2}dx}}>0$;

(U₄) 

$V\in {\mathcal{C}}^1({\mathbb{R}}^2,\mathbb{R})$, there exists ${\beta }_0\ge 0$ such that $V\left(x\right)+{\displaystyle \frac{1}{2}}⟨\nabla V\left(x\right),x⟩\ge {\beta }_0$.

System (49) has a nontrivial solution if one of the following conditions holds:(i) 

$3<\mu <4$, ${\beta }_0=0$;

(ii) 

$\mu =3$, ${\beta }_0>0$;

(iii) 

$2<\mu <3$, $0<\omega <{\displaystyle \frac{\sqrt{{\beta }_0(\mu -2)(4-\mu )}}{3-\mu }}$, ${\beta }_0>0$.

If $\mu =4$, the authors gained the existence of nontrivial solutions for the system (49) under the conditions $\left(U_3\right)$, $\left(R_4\right)-\left(R_7\right)$, $\left(AR\right)$ and ${\beta }_0=0$. Assuming that $F\left(t\right)\ge 0$ for $t\in \mathbb{R}$, f satisfies $\left(R_4\right)-\left(R_7\right)$ and $\left(F_5\right)$, and V satisfies (U₅) 

$V\in \mathcal{C}({\mathbb{R}}^2,\mathbb{R})$, ${inf}_{u\in H^1\left({\mathbb{R}}^2\right)\setminus \left\{0\right\}}{\displaystyle \frac{{\int }_{{\mathbb{R}}^2}\left({|\nabla u|}^2+V\left(x\right)u^2\right)dx}{{\int }_{{\mathbb{R}}^2}{u}^{2}dx}}>0$, and there exists ${\nu }_0>0$ such that

$\underset{\left|y\right|\to \infty }{lim}\mathrm{meas}\left(\{x\in {\mathbb{R}}^2:|x-y|\le {\nu }_0,V\left(x\right)\le M\}\right)=0\mathrm{for}\mathrm{any}M>0,$

system (49) has a nontrivial solution. Secondly, ref. [53] studied the existence of solutions for system (49) under subcritical growth. When f satisfies $\left(R_5\right),\left(AR\right)$ and ${\left(R_4\right)}^{\prime }$ (R₄)′ 

for all $\alpha >0$, ${lim}_{\left|t\right|\to \infty }{\displaystyle \frac{\left|f\right(t\left)\right|}{e^{\alpha t^2}}}=0$,

the above conclusions hold true. Finally, ref. [53] studied the nonexistence of solutions for system (49) under $V\left(x\right)=1$. When f satisfies (R₈) 

$f\left(0\right)=0$, there exists ${\vartheta }_0>1$ such that $f\left(t\right)t+({\vartheta }_0-1)t^2\le 2{\vartheta }_{0}F(x,t)$,

system (49) only has trivial solution. Later, Wen and Jin in [54] generalized $\left(R_7\right)$ to ${\left(R_7\right)}^{\prime }$

(R₇)′ 

${lim\; inf}_{\left|t\right|\to \infty }{\displaystyle \frac{t^{2}F\left(t\right)}{e^{{\alpha }_0{t}^2}}}\ge \kappa >{\displaystyle \frac{2}{e{\alpha }_0^2}}{min}_{r>0}{r}^{-2}{e}^{{\displaystyle \frac{1}{2}}{r}^2(V_r+{\omega }^2)}$, where $V_r:={max}_{\left|x\right|\le r}V\left(x\right)$.

The main result in [54] can be stated as follows.

Theorem 9.

Assume that f satisfies $\left(R_4\right)-\left(R_6\right)$, ${\left(R_7\right)}^{\prime }$ and $\left(AR\right)$; V satisfies $\left(U_3\right)-\left(U_4\right)$. Then system (49) has a nontrivial solution if one of the following conditions holds: (i) 

$3<\mu <4$, ${\beta }_0=0$;

(ii) 

$\mu =3$, ${\beta }_0>0$;

(iii) 

$2<\mu <3$, $0<\omega <{\displaystyle \frac{\sqrt{{\beta }_0(\mu -2)(4-\mu )}}{3-\mu }}$, ${\beta }_0>0$.

If $\mu =4$, system (49) has a nontrivial solution under the conditions $\left(R_4\right)-\left(R_6\right)$, ${\left(R_7\right)}^{\prime }$, $\left(AR\right)$, $\left(U_3\right)$ and ${\beta }_0=0$. Assume that $F\left(t\right)\ge 0$ for $t\in \mathbb{R}$; f satisfies $\left(R_4\right)-\left(R_6\right)$, ${\left(R_7\right)}^{\prime }$ and $\left(F_5\right)$, and V satisfies $\left(U_5\right)$, so system (49) has a nontrivial solution. When $V\left(x\right)=\lambda$, the above conclusions hold true.

Proof. 

First, define a family of energy functionals

$\begin{array}{c}\hfill I_{\lambda }\left(u\right)={\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^2}\left({|\nabla u|}^2+(m^2-{\omega }^2)u^2\right)dx-{\displaystyle \frac{1}{2}}{\int }_{{\mathbb{R}}^2}\omega {\varphi }_u{u}^{2}dx-\lambda {\int }_{{\mathbb{R}}^2}F\left(u\right)dx,\end{array}$

where $\lambda \in [{\displaystyle \frac{1}{2}},1]$. The Pohozave identity for the critical points of I is

$\begin{array}{c}\hfill {\mathcal{P}}_{\lambda }\left(u\right)={\int }_{{\mathbb{R}}^2}\left(V\left(x\right)+{\displaystyle \frac{1}{2}}⟨\nabla V\left(x\right),x⟩\right)u^2-{\int }_{{\mathbb{R}}^2}(2\omega +{\varphi }_u){\varphi }_u{u}^{2}dx-2\lambda {\int }_{{\mathbb{R}}^2}F\left(u\right)dx=0.\end{array}$

Combining the conditions of Theorem 9 and the monotonicity trick, for every $\lambda \in [{\displaystyle \frac{1}{2}},1]$, there exists a bounded ${\left(PS\right)}_{c_{\lambda }}$ sequence $\left\{u_n^{\lambda }\right\}$, where $c_{\lambda }$ is the mountain pass energy level for the functional $I_{\lambda }$. It follows from the property $-\omega \le {\varphi }_u\le 0$ that

$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^2}\omega {\varphi }_u{u}^{2}dx\le {\omega }^2{\parallel u\parallel }_{L^2\left({\mathbb{R}}^2\right)}^2,\end{array}$

Then

(52)$\begin{array}{c}\hfill c_{\lambda }<{\displaystyle \frac{2\pi }{{\alpha }_0}}.\end{array}$

With the fact that $\left\{u_n^{\lambda }\right\}$ is bounded, there exists $u^{\lambda }\in H_r^1\left({\mathbb{R}}^2\right)$ such that $u_n^{\lambda }\rightharpoonup u^{\lambda }$ in $H_r^1\left({\mathbb{R}}^2\right)$. By using (52), $u^{\lambda }\ne 0$. Choose ${\lambda }_n\to 1$; then we have

$I_{{\lambda }_n}\left(u^{{\lambda }_n}\right)\le c_{{\displaystyle \frac{1}{2}}},I_{{\lambda }_n}^{\prime }\left(u^{{\lambda }_n}\right)=0,{\mathcal{P}}_{{\lambda }_n}\left(u^{{\lambda }_n}\right)=0.$

Notice that, for any $v\in H_r^1\left({\mathbb{R}}^3\right)$,

$I\left(u^{{\lambda }_n}\right)=I_{{\lambda }_n}\left(u^{{\lambda }_n}\right)+({\lambda }_n-1){\int }_{{\mathbb{R}}^2}F\left(u^{{\lambda }_n}\right)dx,$

$⟨I^{\prime }\left(u^{{\lambda }_n}\right),v⟩=⟨I_{{\lambda }_n}^{\prime }\left(u^{{\lambda }_n}\right),v⟩+({\lambda }_n-1){\int }_{{\mathbb{R}}^2}f\left(u^{{\lambda }_n}\right)u^{{\lambda }_n}dx.$

Then $\left\{u^{{\lambda }_n}\right\}$ is a ${\left(PS\right)}_{c_1}$ sequence for I. Repeat the above proof; there exists $0\ne u_0\in H_r^1\left({\mathbb{R}}^2\right)$ such that $u^{{\lambda }_n}\rightharpoonup u_0$ in $H_r^1\left({\mathbb{R}}^2\right)$, which implies $u_0$ is a nontrivial solution for system (49). □

6. Discussion

This paper aims to synthesize and analyze research progress on solving various problems in autonomous and non-autonomous Klein–Gordon–Maxwell systems using variational methods through a literature review, providing researchers in the field with a clear research framework and insights for future directions.

Two promising directions for future research are as follows:

(1). Existence of sign-changing solutions under vanishing potentials;

(2). Existence of solutions under critical (BL) conditions.

Author Contributions

Conceptualization, X.-Q.L. and C.-L.T.; Methodology, C.-L.T.; Writing—Original Draft Preparation, X.-Q.L.; Writing—Review and Editing, C.-L.T.; Supervision, C.-L.T. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

No new data were created or analyzed in this study.

Conflicts of Interest

The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Footnotes

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