Infinitely many solutions for nonlinear Klein–Gordon–Maxwell system with general nonlinearity
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Abstract
Keywords
Klein–Gordon–Maxwell system High energy solutions Variational methodsMSC
35J20 35J60 35Q401 Introduction and main result
Since then, a lot of works have been devoted to system (1.1), and we cite a couple of them. For example, Azzollini and Pomponio [1] proved the existence of a ground state solution for the very special power nonlinearity \(f(z)=z^{q2}z\) with \(3\leq q<5\) and \(m_{0}>\omega \), or \(1< p<3\) and \(m_{0} \sqrt{p1}>\omega \sqrt{5p}\). Georgiev and Visciglia [11] also introduced a system like (1.1) with potentials; however, they considered a small external Coulomb potential in the corresponding Lagrangian density. Chen and Tang [7] obtained the existence of two solutions in the radically symmetric function space by using the mountain pass theorem and Ekeland’s variational principle for the nonhomogeneous case. For the critical growth case, that is, \(f(z)=\mu z^{p2}z+z^{4}z\), Cassani [6] showed that the above system has at least a radially symmetric solution when \(4< p<6\) or \(p=4\) provided that \(\mu >0\) is sufficiently large. Soon after that, Carrião et al. [5] also studied the existence of a radially symmetric solution for \(2< p<6\), which extended and generalized the results in [1] and [6], respectively.
 \((V)\)

\(V\in C(\mathbb{R}^{3}, \mathbb{R})\) and \(\inf_{x\in \mathbb{R}^{3}}V(x)>0\);
 \((V_{2})\)
 there exists a constant \(r>0\) such that$$\begin{aligned} \lim_{ \vert y \vert \rightarrow +\infty }\operatorname{meas} \bigl(\bigl\{ x\in \mathbb{R}^{3}: \vert xy \vert \leq r, V(x)\leq M\bigr\} \bigr)=0, \quad \forall M>0; \end{aligned}$$
 \((h_{0})\)

\(g\in C(\mathbb{R}^{3}\times \mathbb{R},\mathbb{R})\), \(g(x,t)t\geq 0\), and \(\lim_{t\rightarrow 0}\frac{g(x,t)}{t}=0\) uniformly in \(x\in \mathbb{R}^{3}\);
 \((h_{1})\)

there exists \(c>0\) such that \(g(x,t)\leq c(1+t^{p1})\) for \(2< p <6\);
 \((h_{2})\)

there exists \(\mu >4\) such that \(\mu G(x,t)\leq g(x,t)t\) for all \((x,t)\in \mathbb{R}^{3}\times \mathbb{R}\), where \(G(x,t)=\int _{0}^{t}g(x,s)\,ds\);
 \((h_{3})\)

there exists \(4<\alpha <6\) such that \(\liminf_{t\rightarrow \infty }\frac{G(x,t)}{t^{\alpha }}>0\) uniformly in \(x\in \mathbb{R}^{3}\);
 \((h_{4})\)

\(\lim_{t\rightarrow \infty }\frac{g(x,t)}{t ^{3}}=\infty \) uniformly in \(x\in \mathbb{R}^{3}\);
 \((h_{5})\)

\(\widetilde{G}(x,t)=\frac{1}{4}g(x,t)tG(x,t)\rightarrow \infty \) as \(t\rightarrow \infty \) uniformly in \(x\in \mathbb{R} ^{3}\).
 \((V_{1})\)

\(V\in C(\mathbb{R}^{3}, \mathbb{R})\) and \(\inf_{x\in \mathbb{R}^{3}}V(x)>\infty \);
 \((h_{6})\)

\(\lim_{t\rightarrow \infty }\frac{G(x,t)}{t ^{4}}=\infty \) uniformly in \(x\in \mathbb{R}^{3}\), and \(g(t)t\geq 0\);
 \((h_{7})\)
 there exists \(r>0\) such that$$\begin{aligned} G(x,t)\le \frac{1}{4}g(x,t)t, \quad \forall (x,t)\in \mathbb{R}^{3} \times \mathbb{R}, \vert t \vert \geq r. \end{aligned}$$
 \((g_{0})\)
 \(g\in C(\mathbb{R}^{3}\times \mathbb{R}, \mathbb{R})\), and there exist constants \(c_{1}, c_{2}>0\) and \(p\in (2, 6)\) such that$$\begin{aligned} \bigl\vert g(x,t) \bigr\vert \le c_{1} \vert t \vert +c_{2} \vert t \vert ^{p1},\quad \forall (x,t)\in \mathbb{R}^{3}\times \mathbb{R}; \end{aligned}$$
 \((g_{1})\)

\(\lim_{t\rightarrow \infty }\frac{G(x,t)}{t ^{4}}=\infty \) uniformly in x, and there exists \(r_{0}\ge 0\) such that \(G(x,t)\ge 0\), \(\forall (x, t)\in \mathbb{R}^{3}\times \mathbb{R}\), \(t\ge r_{0}\);
 \((g_{2})\)
 there exist constants β, \(r_{1}\) such that$$\begin{aligned} G(x,t)\le \frac{1}{4}g(x,t)t+\beta t^{2}, \quad \forall (x, t) \in \mathbb{R}^{3}\times \mathbb{R}, \vert t \vert \geq r_{1}; \end{aligned}$$
 \((g_{3})\)

\(g(x,t)=g(x,t)\) for all \((x, t)\in \mathbb{R}^{3} \times \mathbb{R}\).
Now, we are ready to state the main result of this paper as follows.
Theorem 1.1
Remark 1.2
The remainder of the paper is organized as follows. In Sect. 2, we formulate the variational setting for system (1.2) and introduce some useful preliminaries. We prove Theorem 1.1 in Sect. 3.
2 Variational setting and preliminary results
 \((\tilde{V}_{1})\)

\(V\in C(\mathbb{R}^{3}, \mathbb{R})\) and \(\inf_{x\in \mathbb{R}^{3}}V(x)>0\).
In order to reduce functional (2.2), we need the following technical result (see [12]).
Lemma 2.1
 (i)
\(\omega \leq \phi _{z}\leq 0\)on the set\(\{xz(x) \neq 0\}\);
 (ii)
\(\\phi _{z}\_{\mathcal{D}^{1,2}}\leq C_{0}\z\^{2}\), and\(\int _{\mathbb{R}^{3}}\phi _{z}z^{2}\,dx\leq C_{0}\z\^{4}_{12/5} \leq C_{0}\z\^{4}\).
Now the functional Φ obtained is not strongly indefinite anymore, and we will look for its critical points since if the pair \((z,\phi ) \in E\times \mathcal{D}^{1,2}(\mathbb{R}^{3})\) is a critical point for \(\mathcal{J}\), then z is a critical point for Φ with \(\phi =\phi _{z}\). Recall that a sequence \(\{z_{n}\}\subset E\) is said to be a Cerami sequence (\((C)_{c}\)sequence in short) if \(\varPhi (z_{n}) \rightarrow c\) and \((1+\z_{n})\\varPhi '(z_{n})\rightarrow 0\), Φ is said to satisfy the Cerami condition (\((C)_{c}\)condition in short) if any \((C)_{c}\)sequence has a convergent subsequence.
In order to obtain the existence of high energy solutions, we will use the symmetric mountain pass theorem of Rabinowitz [14]. It should be noted that the symmetric mountain pass theorem is established under the Palais–Smale condition. Since the deformation lemma is still valid under the \((C)_{c}\)condition, we see that the symmetric mountain pass theorem also holds under the \((C)_{c}\)condition.
Proposition 2.2
([14])
 \((I_{1})\)

\(\varphi (0)=0\), \(\varphi (u)=\varphi (u)\)for all\(u\in X\);
 \((I_{2})\)

there exist constants\(\rho , \alpha >0\)such that\(\varphi _{\partial B_{\rho }\cap Z}\ge \alpha \);
 \((I_{3})\)

for any finite dimensional subspace\(\tilde{X} \subset X\), there is\(R=R(\tilde{X})>0\)such that\(\varphi (u) \le 0\)on\(\tilde{X}\setminus B_{R}\);
Lemma 2.3
Lemma 2.4
Assume that\((\tilde{V}_{1})\), \((V_{2})\), and\((g_{0})\)hold, there exists a positive constantρsuch that\(\varPhi _{\partial B_{\rho }\cap Z_{m}}>0\).
Proof
Lemma 2.5
Assume that\((\tilde{V}_{1})\), \((V_{2})\), \((g_{0})\), and\((g_{1})\)hold; for any finite dimensional subspace\(\tilde{E}\subset E\), there is\(R=R(\tilde{E})>0\)such that\(\varPhi (z) \le 0\)for any\(z\in \tilde{E}\backslash B_{R}\).
Proof
Now we discuss the property of the \((C)_{c}\)sequence, we have the following lemma.
Lemma 2.6
Assume that\((\tilde{V}_{1})\), \((V_{2})\), \((g_{0})\)–\((g_{2})\)hold. Then any\((C)_{c}\)sequence ofΦis bounded.
Proof
Lemma 2.7
Assume that\((\tilde{V}_{1})\), \((V_{2})\), and\((g_{0})\)–\((g_{2})\)hold. ThenΦsatisfies the\((C)_{c}\)condition.
Proof
3 Proof of the theorem
In this section, we give the proof of Theorem 1.1.
Proof of Theorem 1.1
Obviously, \((g_{3})\) implies that \(\varPhi (0)=0\) and Φ is even. Lemmas 2.4 and 2.5 imply that Φ satisfies the geometry structure of Proposition 2.2. Lemmas 2.6 and 2.7 show that Φ satisfies the \((C)_{c}\)condition. Thus, by Proposition 2.2, system (2.1) possesses a sequence of infinitely many nontrivial solutions \(\{z_{n}\}\) such that \(\varPhi (z_{n})\rightarrow \infty \) as \(n\rightarrow \infty \). Moreover, system (1.2) also possesses a sequence of infinitely many nontrivial solutions \(\{z_{n}\}\) such that \(\varPhi (z_{n})\rightarrow \infty \) as \(n\rightarrow \infty \). □
Notes
Acknowledgements
The authors are grateful for the referee’s helpful suggestions and comments.
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Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Funding
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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