Nonexistence and existence of nontrivial solutions for Klein–Gordon–Maxwell systems with competing nonlinearities
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Abstract
Keywords
Klein–Gordon–Maxwell system Competing nonlinearities Ekeland’s variational principle Mountain pass theorem Variational methodsMSC
35J50 35B38 35D301 Introduction and main results
 (a)
\(a\in L^{\frac{6}{6r}}(\mathbb{R}^{3})\) is a positive potential function.
 (b)
\(b\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{3})\) is a positive potential function.
 (c)
\(a(\frac{a}{b})^{\frac{r2}{qr}}\in L^{\frac{3}{2}}( \mathbb{R}^{3})\).
 (i)
\(m_{0}>\omega >0\) and \(p\in (4,6)\);
 (ii)
\(m_{0}\sqrt{\frac{p2}{2}}>\omega >0\) and \(p\in (2,4)\).
As far as we know, there are few results about Klein–Gordon–Maxwell systems with similar competing nonlinearities to our system \((\mathrm{P})\). Generally, system \((\mathrm{P})\) can be transformed into a single equation with a nonlocal term by dual methods (see Sect. 2). But in contrast to the problems with a purely single equation (similar to [25, 26, 33]), the nonlocal term brings about some difficulties to us. Firstly, the functional associated with system \((\mathrm{P})\) is no longer weakly lower semicontinuous, which is very important to get the global minimizer in [26, 28, 33]. Secondly, since we lack the compact embedding \(H^{1}(\mathbb{R}^{3})\hookrightarrow \hookrightarrow L^{p}(\mathbb{R}^{3})\), \(p\in (2,6)\), it increases difficulty in verifying that the functional associated with system \((\mathrm{P})\) satisfies the Palais–Smale condition.
The main result of our paper reads as follows.
Theorem 1.1
 (i)
system\((\mathrm{P})\)has only the trivial solution for\(\lambda < \lambda _{0}\);
 (ii)
system\((\mathrm{P})\)has at least two weak nontrivial solutions for\(\lambda >\lambda ^{*}\).
Remark 1.1
Our assumption (b) is the same condition as in [28, 33]. Our assumption (c) is (1.5) for \(N=3\), \(P=2\), \(s=1\). Different from (1.3) or (1.4), the local integrability hypothesis of a is not necessary in our paper.
Remark 1.2
If \(r<2\), for every \(\lambda >0\), \(q>1\), it is easy to prove that the functional associated with system \((\mathrm{P})\) is coercive. Similar to our Lemma 3.2, the functional associated with system \((\mathrm{P})\) also satisfies the Palais–Smale condition. Then the functional has a global minimizer via Ekeland’s variational principle and a sequence of solutions with negative energy decreasing to zero via Clark’s theorem.
Throughout the paper, we denote by C various positive constants, whose value may be different from line to line and is not essential to the problem.
2 Preliminary
In this section, we give some preliminary results which will be used to prove our main results.
First of all, we establish the variational framework for system \((\mathrm{P})\).
Moreover, the function \(\phi _{u}\) has the following properties.
Lemma 2.1
 (i)
\(\omega \leq \phi _{u}\leq 0\)on the set\(\{x\vert u(x) \neq 0\}\);
 (ii)There exist positive constants\(C_{1}\), \(C_{2}\)such that$$ \Vert \phi _{u} \Vert _{D^{1,2}}\leq C_{1} \Vert u \Vert ^{2} \quad \textit{and} \quad \int _{\mathbb{R}^{3}} \vert \phi _{u} \vert u^{2} \,dx\leq C_{2} \Vert u \Vert ^{4}. $$
Definition 2.1
Let X be a Banach space, we say that functional \(I\in C^{1}(X, \mathbb{R})\) satisfies the Palais–Smale condition at the level \(c\in \mathbb{R}\) ((PS)_{c} in short) if any sequence \(\{u_{n}\} \subset X \) satisfying \(I(u_{n}) \to c\), \(I'(u_{n}) \to 0\) as \(n\to \infty \), has a convergent subsequence. I satisfies the (PS) condition if I satisfies the (PS)_{c} condition at any \(c\in \mathbb{R}\).
In order to get the global minimizer, we need the famous Ekeland variational principle.
Lemma 2.2
(Ekeland’s variational principle, [35])
In order to get the second nontrivial solution, we need a modification of mountain pass theorem.
Lemma 2.3
(Theorem A.3 in [26])
3 Proof of the main results
In this section we will prove our main results. Firstly, in the same spirit of the proof of Lemma 3.1 in [33], we can also get the functional I is coercive. And since \(a\in L^{\frac{6}{6r}}( \mathbb{R}^{3})\) implies \(a^{\frac{6}{6r}}\in L^{1}_{\mathrm{loc}}(\mathbb{R} ^{3})\), the assumption \(a\in L_{\mathrm{loc}}^{1}(\mathbb{R}^{3})\) is not necessary in this proof.
Lemma 3.1
(Lemma 3.1 in [33])
Assume that (a)–(c) hold, the functionalIis coercive and bounded from below in E.
In contrast to [33], problem \((\mathrm{P}')\) contains a nonlocal term \(\phi _{u}\). It brings about some difficulties to us. Since we lack the compact embedding \(H^{1}(\mathbb{R}^{3})\hookrightarrow \hookrightarrow L^{p}(\mathbb{R}^{3})\), \(p\in (2,6)\), it increases difficulty in verifying that the functional satisfies the Palais–Smale condition. Here we prove that the functional satisfies the Palais–Smale condition under the integrability assumptions on a and b.
Lemma 3.2
Assume that (a)–(c) hold, the functionalIsatisfies the Palais–Smale condition.
Proof

\(u_{n}\rightharpoonup u\) in \(H^{1}(\mathbb{R}^{3})\), \(u_{n}\rightharpoonup u\) in E;

\(u_{n}\rightarrow u\) in \(L_{\mathrm{loc}}^{s}(\mathbb{R}^{3})\), where \(s\in [2,6)\); \(u_{n}(x)\rightarrow u(x)\) a.e. in \(\mathbb{R}^{3}\);

and \(\vert u_{n}\vert ^{r}\rightharpoonup \vert u\vert ^{r}\), \(\vert u_{n}\vert ^{r2}u_{n}u \rightharpoonup \vert u\vert ^{r}\), \(\vert u\vert ^{r2}uu_{n}\rightharpoonup \vert u\vert ^{r}\) in \(L^{\frac{6}{r}}(\mathbb{R}^{3})\).
Though the functional I is no longer weakly lower semicontinuous, which is very important to get the global minimizer in [26, 28, 33], we can still prove that I still has a global minimizer in E via Ekeland’s variational principle.
Lemma 3.3
There exists\(\lambda ^{*}>0\)such thatIenjoys a global minimizer\(u_{\lambda }^{1}\in E\)with\(I(u_{\lambda }^{1})<0\)for every\(\lambda >\lambda ^{*}\).
Proof
At last, similar to [33], we will prove that, for every \(\lambda >\lambda ^{*}\), system \((\mathrm{P})\) has a second weak solution \(u_{\lambda }^{2}\) via a modification of the mountain pass theorem of Ambrosetti and Rabinowitz. By choosing \(X=E\), \(Y=H^{1}(\mathbb{R}^{3})\), we have the following.
Lemma 3.4
Proof
Lemma 3.5
If (a)–(c) hold, then system\((\mathrm{P})\)enjoys a nontrivial solution\(u_{\lambda }^{2}\in E\)with\(I(u_{\lambda }^{2})=c_{\lambda }>0\)for every\(\lambda >\lambda ^{*}\).
Proof
Lemma 3.3 implies that there exists a global nontrivial minimizer \(u_{\lambda }^{1}\in E\) of I with \(I(u_{\lambda }^{1})<0\) for every \(\lambda >\lambda ^{*}\). By choosing \(e=u_{\lambda }^{1}\) in Lemma 3.4, I satisfies the geometrical structure of Lemma 2.3. Thus I has a (PS)\(_{c_{\lambda }}\) sequence for every \(\lambda >\lambda ^{*}\). From Lemma 3.2, we can get that I enjoys a nontrivial solution \(u_{\lambda }^{2}\) with \(I(u_{\lambda }^{2})=c_{\lambda }>0>I(u_{ \lambda }^{1})\) for every fixed \(\lambda \in (\lambda ^{*},+\infty )\) (for more details, see the proof of Theorem A.3 in [26]). □
Proof of Theorem 1.1
(ii) Combine Lemma 3.3 with Lemma 3.5, system \((\mathrm{P})\) enjoys at least two nontrivial solutions for every \(\lambda >\lambda ^{*}\). □
Notes
Acknowledgements
Not applicable.
Availability of data and materials
Not applicable.
Authors’ contributions
All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.
Funding
This work is supported by the National Natural Science Foundation of China (Grant No. 11701346, 11571209, 11671239).
Competing interests
The authors declare that they have no competing interests.
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