Open Access

Multiplicity of small negative-energy solutions for a class of semilinear elliptic systems

Boundary Value Problems20162016:107

https://doi.org/10.1186/s13661-016-0616-5

Received: 25 December 2015

Accepted: 20 May 2016

Published: 2 June 2016

Abstract

This paper is concerned with the following semilinear elliptic systems:
$$\left \{ \textstyle\begin{array}{@{}l} -\Delta u+V(x)u=H(x)F_{u}(x, u, v), \quad x\in\mathbb{R}^{N},\\ -\Delta v+V(x)v=H(x)F_{v}(x, u, v), \quad x\in\mathbb{R}^{N},\\ u(x)\rightarrow0,\qquad v(x)\rightarrow0\quad \mbox{as } |x|\rightarrow\infty, \end{array}\displaystyle \right . $$
where \(V(x)\), \(H(x)\) are nonnegative continuous functions. Under some appropriate assumptions on \(V(x)\), \(H(x)\), and \(F(x, u, v)\), we prove the existence of infinitely many small negative-energy solutions by using the fountain theorem established by Zou. Recent results from the literature are extended.

Keywords

semilinear elliptic systems multiple solutions variant fountain theorem variational methods

MSC

35B38 35J20

1 Introduction

In this paper, we consider the existence and multiplicity of solutions to the following semilinear elliptic systems:
$$ \left \{ \textstyle\begin{array}{@{}l} -\Delta u+V(x)u=H(x)F_{u}(x, u, v),\quad x\in\mathbb{R}^{N},\\ -\Delta v+V(x)v=H(x)F_{v}(x, u, v),\quad x\in\mathbb{R}^{N},\\ u(x)\rightarrow0, \qquad v(x)\rightarrow0\quad \mbox{as } |x|\rightarrow\infty, \end{array}\displaystyle \right . $$
(1.1)
where \(V(x)\), \(H(x)\) are nonnegative continuous functions, we assume that the functions \(V(x)\), \(H(x)\), and \(F(x,u,v)\) satisfy the following hypotheses:
(H1): 

\(V\in C(\mathbb{R}^{N},\mathbb {R} )\) satisfies \(\inf_{x\in\mathbb{R}^{N}}V(x)\geq a_{0}>0\), where \(a_{0}>0\) is a constant. Moreover, for any \(M>0\), \(\operatorname{meas} \{x\in\mathbb{R}^{N}: V(x)\leq M\}<\infty\), where meas denotes the Lebesgue measure in \(\mathbb{R}^{N}\).

(H2): 

\(F\in C^{1}(\mathbb {R}^{N}\times\mathbb {R}^{2}, \mathbb {R})\), \(|F_{u}(x,u,v)|\leq c(|(u,v)|+|(u,v)|^{p-1})\), and \(|F_{v}(x,u,v)|\leq c(|(u,v)|+|(u,v)|^{q-1})\) for some \(1< p, q<2\), where c is a positive constant, and \(|(u,v)|=(u^{2}+v^{2})^{\frac{1}{2}}\).

(H3): 

\(F(x, 0,0)=0\), \(F(x, u, v)\geq0\) for all \((x,u,v)\in\mathbb {R}^{N}\times\mathbb {R}^{2}\), and for some \(1<\mu <2\), there exists \(c_{1}>0\) such that \(F(x, u, v)\geq c_{1}|(u,v)|^{\mu }\).

(H4): 

\(H(x)\geq0\) and \(H(x)\in L^{\frac{2}{2-p}}(\mathbb {R}^{N}, \mathbb{R})\cap L^{\frac{2}{2-q}}(\mathbb {R}^{N}, \mathbb{R}) \cap L^{\frac{2}{2-\mu}}(\mathbb {R}^{N}, \mathbb{R})\cap L^{\infty }(\mathbb {R}^{N}, \mathbb{R}) \).

(H5): 

\(F(x,u,v)=F(x,-u,-v)\) for all \((x,u,v)\in\mathbb {R}^{N}\times \mathbb {R}^{2}\).

When Ω is a bounded domain of \(\mathbb{R}^{N}\), the problem
$$ \left \{ \textstyle\begin{array}{@{}l} -\Delta u=\lambda(a(x)u+b(x)v)+F_{u}(x, u, v) \quad\mbox{in }\Omega,\\ -\Delta v=\lambda(b(x)u+c(x)v)+F_{v}(x, u, v) \quad\mbox{in } \Omega,\\ u(x)=v(x)= 0 \quad\mbox{on }\partial\Omega, \end{array}\displaystyle \right . $$
(1.2)
which is related to reaction-diffusion systems that appear in chemical and biological phenomena, including the steady and unsteady state situation (see [14]), has been extensively investigated in recent years. For the results on existence, multiple solutions, and positive solutions to problem (1.2), we refer the readers to [1, 49] and the references therein. Qu and Tang [5] obtained the existence and multiplicity of weak solutions of problem (1.2) by using the Ekeland variational principle, the mountain pass theorem, and the saddle point theorem in critical point theory, and by applying the local linking theorem and the saddle point theorem some new existence theorems of weak solutions were obtained by Duan et al. [6]. In [7], by using Morse theory the multiplicity of solutions was obtained for cooperative elliptic systems at resonance. Costa and Magalhães [8, 9] researched subquadratic perturbation problems of semiliner elliptic systems by minimax methods.
Recently, the problems in the whole space \(\mathbb {R}^{N}\) were considered in some works. For example, see [1015] and the references therein. Cao and Tang [10] studied the following Schrödinger systems:
$$ \left \{ \textstyle\begin{array}{@{}ll} -\Delta u+V(x)u=F_{u}(x, u, v), \quad x\in\mathbb{R}^{N},\\ -\Delta v+V(x)v=F_{v}(x, u, v), \quad x\in\mathbb{R}^{N}. \end{array}\displaystyle \right . $$
(1.3)
Under suitable assumptions on \(F(x,u,v)\), they obtained the existence of infinitely many solutions characterized by the number of nodes of each component. When \(N\geq3\), \(V(x)=0\), \(F_{u}(x, u, v)=p(x)f(v)\), and \(F_{v}(x, u, v)=q(x)g(u)\), Zhang et al. [11] obtained the existence and nonexistence of entire solutions to (1.3). Wu [12] obtained five new critical point theorems on the product spaces and studied three existence theorems for the sequence of high-energy solutions to problem (1.3), whereas Zhou et al. [13] established the existence of high-energy solutions to (1.3) under some conditions that are weaker than those in [12], which unify and sharply improve the recent results in [14].

Inspired by all these facts, the aim of this paper is to study the multiplicity of small negative-energy solutions to problem (1.1) via variational methods, which have been widely used to study Schrödinger equations; see [1623] and the references therein. To the best of our knowledge, there has been few works concerning this case up to now.

Now, we state our main results.

Theorem 1.1

Suppose that conditions (H1)-(H5) hold. Then problem (1.1) possesses infinitely many solutions \(\{(u_{k},v_{k})\}\) satisfying
$$\begin{aligned} &\frac{1}{2} \int_{\mathbb{R}^{N}}\bigl(|\nabla u_{k}|^{2}+V(x)u_{k}^{2} \bigr)\,dx+ \frac{1}{2} \int_{\mathbb{R}^{N}}\bigl(|\nabla v_{k}|^{2}+V(x)v_{k}^{2} \bigr)\,dx\\ &\quad{}- \int_{\mathbb{R}^{N}}H(x)F(x, u_{k}, v_{k})\,dx \rightarrow0^{-} \quad \textit{as } k\rightarrow\infty. \end{aligned}$$

The remainder of this paper is as follows. In Section 2, we present some preliminary results. In Section 3, we give a proof of the main result.

2 Variational setting and preliminaries

In this section, we outline the variational framework for problem (1.1) and give some preliminary lemmas.

Let
$$H^{1}\bigl(\mathbb{R}^{N}\bigr)=\bigl\{ u\in L^{2} \bigl(\mathbb{R}^{N}\bigr): \nabla u \in L^{2}\bigl( \mathbb{R}^{N}\bigr) \bigr\} $$
with the norm
$$\|u\|_{H^{1}}=\biggl( \int_{\mathbb{R}^{N}}\bigl(|\nabla u|^{2}+|u|^{2}\bigr) \,dx\biggr)^{\frac{1}{2}}. $$
Let
$$X=\biggl\{ u\in H^{1}\bigl(\mathbb{R}^{N}\bigr)\Big| \int_{\mathbb{R}^{N}}\bigl(|\nabla u|^{2}+V(x)u^{2} \bigr)\,dx< +\infty\biggr\} $$
with the inner product and norm
$$\langle u,v\rangle_{X}= \int_{\mathbb{R}^{N}}\bigl(\nabla u\nabla v+V(x)uv\bigr)\,dx,\quad \|u \|_{X}=\langle u, u\rangle_{X}^{\frac{1}{2}}. $$
As usual, for \(1\leq p<+\infty\), we let
$$\|u\|_{p}=\biggl( \int_{\mathbb{R}^{N}}\bigl|u(x)\bigr|^{p}\,dx\biggr)^{\frac{1}{p}},\quad u\in L^{p}\bigl(\mathbb{R}^{N}\bigr), $$
and
$$\|u\|_{\infty}=\operatorname{ess} \sup_{x\in\mathbb{R}^{N}}\bigl|u(x)\bigr|,\quad u\in L^{\infty}\bigl(\mathbb{R}^{N}\bigr). $$
Then \(E=X\times X\) is a Hilbert space with the inner product
$$\bigl\langle (u,v),(\varphi,\psi)\bigr\rangle =\langle u,\varphi \rangle_{X}+\langle v,\psi\rangle_{X},\quad (u,v), (\varphi,\psi) \in X\times X, $$
and the norm
$$\bigl\| (u,v)\bigr\| ^{2}=\bigl\langle (u,v),(u,v)\bigr\rangle =\|u \|_{X}^{2}+\|v\| _{X}^{2},\quad (u,v), ( \varphi,\psi)\in X\times X. $$
Define the functional I on E by
$$ I(u,v)=\frac{1}{2}\bigl\| (u,v)\bigr\| ^{2}- \int_{\mathbb{R}^{N}}H(x)F(x, u, v)\,dx. $$
(2.1)
Then a weak solution of system (1.1) is a critical point of I if I is continuously differentiable on E.

Moreover, we have the following compactness lemma.

Lemma 2.1

Under assumption (H1), the embedding \(E\hookrightarrow L^{r}(\mathbb{R}^{N})\times L^{r}(\mathbb{R}^{N})\) is continuous for \(2\leq r\leq2^{*}\), and \(E\hookrightarrow L^{r}(\mathbb {R}^{N})\times L^{r}(\mathbb{R}^{N})\) is compact for \(2\leq r<2^{*}\).

Proof

By Lemma 3.4 in [24] we know that, under assumption (H1), the embedding \(X\hookrightarrow L^{r}(\mathbb{R}^{N})\) is continuous for \(r\in[2, 2^{*}]\) and that \(X\hookrightarrow L^{r}(\mathbb{R}^{N})\) is compact for \(r\in[2, 2^{*})\), that is, there exist constants \(C_{r}>0\) such that \(\|u\|_{r}\leq C_{r}\|u\|_{X}\), \(\forall u\in X\), and for any bounded sequence \(\{u_{n}\}\subset X\), there exists a subsequence of \(\{u_{n}\} \) such that \(u_{n}\rightarrow u_{0}\) in \(L^{r}(\mathbb{R}^{N})\), \(r\in [2, 2^{*})\). Therefore, for any \((u,v)\in E\), there exists \(C>0\) such that
$$\bigl\| (u,v)\bigr\| _{r}^{r}\leq C\bigl(\|u\|_{r}^{r}+ \|u\|_{r}^{r}\bigr)\leq C\bigl(\|u\|_{X}^{r}+ \|u\|_{X}^{r}\bigr)\leq C\bigl\| (u,v)\bigr\| ^{r}, $$
that is, \(\|(u,v)\|_{r}\leq C\|(u,v)\|\), so that \(E\hookrightarrow L^{r}(\mathbb{R}^{N})\times L^{r}(\mathbb{R}^{N})\) is continuous for \(2\leq r\leq2^{*}\). On the other hand, suppose that \(\{(u_{n},v_{n})\}\subset E\) are bounded, that is, \(\{u_{n}\}\) and \(\{v_{n}\}\) are bounded in X, then there exist subsequences \(\{u_{n}\}\) and \(\{v_{n}\}\) such that
$$u_{n}\rightarrow u_{0},\qquad v_{n}\rightarrow v_{0} \quad\mbox{in } L^{r}\bigl(\mathbb{R}^{N}\bigr), r\in\bigl[2, 2^{*}\bigr). $$
Therefore,
$$0\leq\|u_{n}-u_{0}\|_{r}^{r}+ \|v_{n}-v_{0}\|_{r}^{r}\leq \bigl\| (u_{n},v_{n})-(u_{0},v_{0}) \bigr\| _{r}^{r}\leq C \bigl(\|u_{n}-u_{0} \|_{r}^{r}+\|v_{n}-v_{0} \|_{r}^{r}\bigr)\rightarrow0 $$
as \(n\rightarrow\infty\), that is,
$$(u_{n},v_{n})\rightarrow(u_{0},v_{0})\quad \mbox{in }L^{r}\bigl(\mathbb{R}^{N}\bigr)\times L^{r} \bigl(\mathbb{R}^{N}\bigr), r\in\bigl[2, 2^{*}\bigr), $$
so that \(E\hookrightarrow L^{r}(\mathbb{R}^{N})\times L^{r}(\mathbb{R}^{N})\) is compact for \(r\in[2, 2^{*})\). The proof is complete. □

Lemma 2.2

If assumptions (H1)-(H2) hold, then \(I\in C^{1}(E, R)\),
$$ \begin{aligned}[b] \bigl\langle I'(u,v), (\varphi,\psi)\bigr\rangle ={}& \int_{\mathbb{R}^{N}}\nabla u\nabla\varphi \,dx + \int_{\mathbb{R}^{N}}V(x)u\varphi \,dx- \int_{\mathbb {R}^{N}}H(x)F_{u}(x,u,v)\varphi \,dx\\ &{} + \int_{\mathbb{R}^{N}}\nabla v\nabla\psi \,dx + \int_{\mathbb{R}^{N}}V(x)v\psi \,dx- \int_{\mathbb {R}^{N}}H(x)F_{v}(x,u,v)\psi \,dx, \end{aligned} $$
(2.2)
and \(\Psi': E\rightarrow E^{*}\) is compact, where \(\Psi(u,v)=\int _{\mathbb{R}^{N}}H(x)F(x,u,v)\,dx\).

Proof

The proof is similar to that of Lemma 3.1 in [12]; we omit it. □

To complete the proof of our theorem, the following theorem will be needed in our argument. Let E be a Banach space with norm \(\|\cdot\|\) and \(E=\overline {\bigoplus_{j\in N}X_{j}}\) with \(\operatorname{dim} X_{j}<\infty\) for any \(j\in N\). Set \(Y_{k}=\bigoplus_{j=0}^{k}X_{j}\), \(Z_{k}=\overline{\bigoplus_{j=k+1}^{\infty}X_{j}}\). Consider the \(C^{1}\) functional
$$\varphi_{\lambda}(u)=A(u)-\lambda B(u),\quad \lambda\in[1,2], $$
where \(A, B: E\rightarrow\mathbb{R}\) are two functionals.

Theorem 2.1

([25], Theorem 2.1)

Suppose that the functional \(\varphi_{\lambda}(u)\) satisfies:
(C1): 

\(\varphi_{\lambda}(u)\) maps bounded sets to bounded sets uniformly for \(\lambda\in[1,2]\). Furthermore, \(\varphi_{\lambda}(-u)=\varphi_{\lambda}(u)\) for all \((\lambda, u)\in [1,2]\times E\).

(C2): 

\(B(u)\geq0\); \(B(u)\rightarrow\infty\) as \(\|u\|\rightarrow \infty\) on any finite-dimensional subspace of E.

(C3): 
There exists \(\rho_{k}>r_{k}>0\) such that
$$\begin{aligned}& a_{k}(\lambda) :=\inf_{u\in Z_{k}, \|u\|=\rho_{k}}\varphi_{\lambda}(u) \geq 0>b_{k}(\lambda):=\max_{u\in Y_{k}, \|u\|=r_{k}}\varphi _{\lambda}(u),\quad \lambda\in[1,2],\\& d_{k}(\lambda):=\inf_{u\in Z_{k}, \|u\|\leq\rho_{k}}\varphi _{\lambda}(u)\rightarrow0\quad \textit{as }k\rightarrow \infty\textit{ uniformly for }\lambda \in[1,2]. \end{aligned}$$
Then there exist \(\lambda_{n}\rightarrow1\) and \(u(\lambda_{n})\in Y_{n}\) such that
$$\varphi_{\lambda_{n}}'|_{Y_{n}}\bigl(u( \lambda_{n})\bigr)=0,\qquad \varphi_{\lambda _{n}}\bigl(u( \lambda_{n})\bigr) \rightarrow c_{k}\in\bigl[d_{k}(2),b_{k}(1) \bigr] \quad\textit{as }n\rightarrow\infty. $$
In particular, if \(\{u(\lambda_{n})\}\) has a convergent subsequence for every k, then \(\varphi_{1}\) has many nontrivial critical points \(\{u_{k}\}\subset E\setminus\{0\}\) satisfying \(\varphi _{1}(u_{k})\rightarrow0^{-}\) as \(n\rightarrow\infty\).

3 Proof of the main result

In order to apply Theorem 2.1 to prove our main result, we define A, B, and \(\varphi_{\lambda}\) on our working space E by
$$ A(u,v)=\frac{1}{2}\bigl\| (u,v)\bigr\| ^{2},\qquad B(u,v)= \int_{\mathbb{R}^{N}}H(x)F(x, u, v)\,dx, $$
(3.1)
and
$$ \varphi_{\lambda}(u,v)=A(u,v)-\lambda B(u,v)=\frac{1}{2}\bigl\| (u,v)\bigr\| ^{2}-\lambda \int_{\mathbb{R}^{N}}H(x)F(x, u, v)\,dx $$
(3.2)
for all \((u,v)\in E\) and \(\lambda\in[1,2]\). Obviously, \(\varphi _{\lambda}(u,v)\in C^{1}(E, R)\) for all \(\lambda\in[1,2]\). We choose a completely orthonormal basis \(\{e_{j}: j\in N\}\) of X and let \(X_{j}=\operatorname{span}\{e_{j}\}\) for all \(j \in N\). Then \(Y_{k}=\operatorname{span} \{e_{1},\ldots,e_{k}\}\), \(Z_{k}=Y_{k}^{\bot }\), and \(E=(Y_{k}\times Y_{k})\oplus(Z_{k}\times Z_{k})\). Note that \(\varphi_{1}=I\), where I is defined in (2.1).

Lemma 3.1

Suppose that conditions (H1), (H3), and (H4) hold. Then \(B(u,v)\geq0\). Furthermore, \(B(u,v)\rightarrow\infty\) as \(\|(u,v)\|\rightarrow\infty\) on any finite-dimensional subspace of E.

Proof

Evidently, by (H3) and (H4), \(B(u,v)=\int _{\mathbb{R}^{N}}H(x)F(x, u, v)\,dx\geq0\), Now we claim that for any finite-dimensional subspace \(\widetilde{E}\subset E\), there exists \(\varepsilon>0\) such that
$$ \operatorname{meas}\bigl\{ x\in\mathbb{R}^{N}: H(x)\bigl|(u,v)\bigr|^{\mu}\geq \varepsilon \bigl\| (u,v)\bigr\| ^{\mu}\bigr\} \geq\varepsilon, \quad\forall(u,v)\in \widetilde{E}. $$
(3.3)
Arguing by contradiction, we assume that there exists a sequence \(\{ (u_{n},v_{n})\}_{n\in\mathbb{N}}\subset\widetilde{E}\setminus\{(0,0)\} \) such that
$$ \operatorname{meas}\biggl\{ x\in\mathbb{R}^{N}: H(x)\bigl|(u_{n},v_{n})\bigr|^{\mu} \geq\frac {1}{n}\bigl\| (u_{n},v_{n})\bigr\| ^{\mu}\biggr\} < \frac{1}{n}. $$
(3.4)
Set \((s_{n},w_{n})=\frac{(u_{n},v_{n})}{\|(u_{n},v_{n})\|}\subset \widetilde{E}\setminus\{(0,0)\}\), then \(\|(s_{n},w_{n})\|=1\) for all \(n\in\mathbb{N}\), and
$$ \operatorname{meas} \biggl\{ x\in\mathbb{R}^{N}: H(x)\bigl|(s_{n},w_{n})\bigr|^{\mu} \geq\frac {1}{n}\biggr\} < \frac{1}{n}. $$
(3.5)
Since \(\operatorname{dim} \widetilde{E}<\infty\), it follows from the compactness of the unit sphere of that there exists a subsequence, say \(\{(s_{n},w_{n})\}\), such that \((s_{n},w_{n})\rightarrow(s_{0},w_{0})\) in . It is easy to verify that \(\|(s_{0},w_{0})\|=1\). In view of the equivalence of the norms on the finite-dimensional space , we have \((s_{n},w_{n})\rightarrow(s_{0},w_{0})\) in \(L^{2}(\mathbb{R}^{N})\), that is,
$$ \int_{\mathbb{R}^{N}}\bigl|(s_{n},w_{n})-(s_{0},w_{0})\bigr|^{2} \,dx\rightarrow0 \quad\mbox{as } n\rightarrow\infty. $$
(3.6)
By (3.6) and the Hölder inequality we have
$$\begin{aligned} &\int_{\mathbb{R}^{N}}H(x)\bigl|(s_{n},w_{n})- (s_{0},w_{0})\bigr|^{\mu}\,dx \\ &\quad\leq\bigl\| H(x) \bigr\| _{{\frac{2}{2-\mu}}}\biggl( \int_{\mathbb {R}^{N}}\bigl|(s_{n},w_{n})- (s_{0},w_{0})\bigr|^{2}\,dx\biggr)^{\frac{\mu}{2}} \rightarrow0 \quad\mbox{as } n\rightarrow\infty. \end{aligned}$$
(3.7)
Therefore, there exist \(\xi_{1}, \xi_{2}>0\) such that
$$ \operatorname{meas} \bigl\{ x\in\mathbb{R}^{N}: H(x)\bigl|(s_{0},w_{0})\bigr|^{\mu} \geq\xi _{1}\bigr\} \geq\xi_{2}. $$
(3.8)
Otherwise, we get
$$ \operatorname{meas} \biggl\{ x\in\mathbb{R}^{N}: H(x)\bigl|(s_{0},w_{0})\bigr|^{\mu} \geq\frac {1}{n}\biggr\} =0, $$
(3.9)
which implies that
$$0\leq \int_{\mathbb{R}^{N}}H(x)\bigl|(s_{0},w_{0})\bigr|^{\mu+2} \,dx \leq\frac{\|(s_{0},w_{0})\|^{2}_{{2}}}{n}\leq\frac {C^{2}\|(s_{0},w_{0})\|^{2}}{n}=\frac{C^{2}}{n}\rightarrow0 \quad\mbox{as } n\rightarrow\infty. $$
Hence, \((s_{0},w_{0})=0\), which contradicts with \(\|(s_{0},w_{0})\|=1\). Therefore, (3.8) holds.
Now let
$$\begin{aligned}& \Omega_{0}=\operatorname{meas} \bigl\{ x\in\mathbb{R}^{N}: H(x)\bigl|(s_{0},w_{0})\bigr|^{\mu }\geq\xi_{1}\bigr\} , \\& \Omega_{n}=\operatorname{meas} \biggl\{ x\in\mathbb{R}^{N}: H(x)\bigl|(s_{0},w_{0})\bigr|^{\mu }< \frac{1}{n}\biggr\} , \\& \Omega_{n}^{c}=\mathbb{R}^{N}\setminus \Omega_{n}=\operatorname{meas} \biggl\{ x\in \mathbb{R}^{N}: H(x)\bigl|(s_{0},w_{0})\bigr|^{\mu}\geq\frac{1}{n} \biggr\} . \end{aligned}$$
Then by (3.5) and (3.8) we get
$$\begin{aligned}[b] \operatorname{meas} (\Omega_{n}\cap \Omega_{0}) &=\operatorname{meas} \bigl(\Omega_{0}\setminus \bigl(\Omega_{n}^{c}\cap\Omega_{0}\bigr)\bigr) \\ &\geq\operatorname{meas} (\Omega_{0})-\operatorname{meas} \bigl( \Omega_{n}^{c}\cap\Omega _{0}\bigr) \\ &\geq\xi_{2}-\frac{1}{n} \end{aligned} $$
for all positive integer n. Let n be large enough such that \(\xi _{2}-\frac{1}{n}\geq\frac{\xi_{2}}{2}\) and \(\frac{\xi_{1}}{2^{\mu}}-\frac{1}{n}\geq\frac{\xi_{1}}{2^{\mu+1}}\). Then we have
$$\begin{aligned}[b] &\int_{\mathbb{R}^{N}}H(x)\bigl|(s_{n},w_{n})-(s_{0},w_{0})\bigr|^{\mu} \,dx \\ &\quad\geq \int_{\Omega_{n}\cap\Omega _{0}}H(x)\bigl|(s_{n},w_{n})-(s_{0},w_{0})\bigr|^{\mu} \,dx\\ &\quad\geq\frac{1}{2^{\mu}} \int_{\Omega_{n}\cap\Omega _{0}}H(x)\bigl|(s_{0},w_{0})\bigr|^{\mu} \,dx- \int_{\Omega_{n}\cap\Omega _{0}}H(x)\bigl|(s_{n},w_{n})\bigr|^{\mu} \,dx \\ &\quad\geq\biggl(\frac{\xi_{1}}{2^{\mu}}-\frac{1}{n}\biggr) \operatorname{meas} ( \Omega_{n}\cap \Omega_{0})>0, \end{aligned} $$
which is a contradiction with (3.7). Therefore, (3.3) holds. For the ε given in (3.3), let
$$\Omega_{(u,v)}=\operatorname{meas} \bigl\{ x\in\mathbb{R}^{N}: H(x)\bigl|(u,v)\bigr|^{\mu}\geq \varepsilon\bigl\| (u,v)\bigr\| ^{\mu}\bigr\} ,\quad \forall(u,v)\in\widetilde{E}\setminus\bigl\{ (0,0)\bigr\} . $$
Then by (3.3)
$$ \operatorname{meas} (\Omega_{(u,v)})\geq\varepsilon,\quad \forall(u,v)\in \widetilde{E}\setminus\bigl\{ (0,0)\bigr\} . $$
(3.10)
Combining (H3) and (3.10), we have
$$B(u,v)\geq \int_{\mathbb{R}^{N}}c_{1}H(x)\bigl|(u,v)\bigr|^{\mu}\,dx\geq \int_{\Omega _{(u,v)}}\varepsilon c_{1}\bigl\| (u,v)\bigr\| ^{\mu} \,dx \geq\varepsilon^{2} c_{1}\bigl\| (u,v)\bigr\| ^{\mu}, $$
which implies that \(B(u,v)\rightarrow\infty\) as \(\|(u,v)\|\rightarrow \infty\) on any finite-dimensional space of E. The proof is completed. □

Lemma 3.2

Suppose that (H1)-(H2) and (H4) are satisfied. Then there exists a sequence \(\rho_{k}\rightarrow0^{+}\) as \(k\rightarrow\infty\) such that \(a_{k}(\lambda) :=\inf_{(u,v)\in Z_{k}\times Z_{k}, \|(u,v)\|=\rho_{k}}\varphi _{\lambda}(u,v)\geq0\) and \(d_{k}(\lambda):=\inf_{(u,v)\in Z_{k}\times Z_{k}, \|(u,v)\|\leq\rho_{k}} \varphi_{\lambda }(u,v)\rightarrow0\) as \(k\rightarrow\infty\) uniformly for \(\lambda\in [1,2]\), where \(Z_{k}=\overline{\bigoplus_{j=k+1}^{\infty }X_{j}}=\overline{\operatorname{span}\{e_{k},\ldots\}}\) for all \(k\in N\).

Proof

Let
$$ \alpha_{k}(r):=\sup_{(u,v)\in Z_{k}\times Z_{k}, \|(u,v)\|=1}\bigl\| (u,v)\bigr\| _{r},\quad \forall k\in N, $$
(3.11)
where \(\|(u,v)\|_{r}=(\int_{\mathbb {R}^{N}}|(u,v)|^{r})^{\frac{1}{r}}\). Then \(\alpha_{k}(r)\rightarrow0\) as \(k\rightarrow\infty\). Indeed, \(\alpha_{k}(r)\) is convergent since \(\alpha_{k}(r)\) are decreasing in k and \(\alpha_{k}(r)\geq0\). Furthermore, for any k, there exists \((u_{k},v_{k})\in Z_{k}\times Z_{k}\) such that \(\|(u_{k},v_{k})\|=1\) and \(\|(u_{k},v_{k})\|_{r}\geq\frac{\alpha_{k}(r)}{2}\).
For any \(\varphi\in X\), \(\varphi=\sum_{n=1}^{\infty}a_{n}e_{n}\), we get
$$\bigl|\langle u_{k},\varphi\rangle_{X}\bigr|= \Biggl| \Biggl\langle u_{k},\sum_{n=k+1}^{\infty}a_{n}e_{n} \Biggr\rangle _{X} \Biggr|\leq\|u_{k}\|_{X} \Biggl\| \sum _{n=k+1} ^{\infty}a_{n}e_{n} \Biggr\| _{X}\leq \Biggl\| \sum_{n=k+1} ^{\infty}a_{n}e_{n} \Biggr\| _{X}\rightarrow0 $$
as \(k\rightarrow\infty\), which implies that \(u_{k}\rightharpoonup0\) in X. Since the embedding \(X\hookrightarrow L^{r}(\mathbb {R}^{N})\) is compact, \(u_{k}\rightarrow0\) in \(L^{r}(\mathbb {R}^{N})\) for \(r\in[2, 2^{*})\). The same argument implies that \(v_{k}\rightarrow0\) in \(L^{r}(\mathbb {R}^{N})\) for \(r\in[2, 2^{*})\). Consequently,
$$\bigl\| (u_{k},v_{k})\bigr\| _{r}^{r} \leq2^{\frac {r}{2}}\bigl(\|u_{k}\|_{r}^{r}+ \|v_{k}\|_{r}^{r}\bigr)\rightarrow \quad\mbox{as } k \rightarrow\infty, $$
that is,
$$ \alpha_{k}(r)\rightarrow0 \quad\mbox{as } k\rightarrow\infty. $$
(3.12)
By (H2) we have
$$ \begin{aligned}[b] \bigl|F(x,u,v)\bigr| &=\bigl|F(x,u,v)-F(x,0,0)\bigr|\\ &\leq \int_{0}^{1}\bigl|F_{u}(x,tu,tv)\bigr||u|\,dt+ \int _{0}^{1}\bigl|F_{v}(x,tu,tv)\bigr||v|\,dt\\ &\leq C\bigl(\bigl|(u,v)\bigr|^{2}+\bigl|(u,v)\bigr|^{p}+\bigl|(u,v)\bigr|^{q} \bigr). \end{aligned} $$
(3.13)
Therefore, by (3.2), (3.11)-(3.13), and the Hölder inequality we get
$$\begin{aligned} \varphi_{\lambda}(u,v) &=\frac{1}{2}\bigl\| (u,v) \bigr\| ^{2}-\lambda \int_{\mathbb{R}^{N}}H(x)F(x, u, v)\,dx \\ &\geq\frac{1}{2}\bigl\| (u,v)\bigr\| ^{2}-2C\bigl(\bigl\| (u,v) \bigr\| _{2}^{2}+\bigl\| H(x)\bigr\| _{\frac {2}{2-p}}\bigl\| (u,v)\bigr\| _{2}^{p}+ \bigl\| H(x)\bigr\| _{\frac{2}{2-q}}\bigl\| (u,v)\bigr\| _{2}^{q}\bigr) \\ &\geq\frac{1}{2}\bigl\| (u,v)\bigr\| ^{2}-2C\bigl(\alpha_{k}^{2}(2) \bigl\| (u,v)\bigr\| ^{2}+\alpha _{k}^{p}(2)\bigl\| (u,v) \bigr\| ^{p}+\alpha_{k}^{q}(2)\bigl\| (u,v)\bigr\| ^{q} \bigr). \end{aligned}$$
(3.14)
By (3.13) there exist a positive integer \(k_{1}\) such that
$$ 2C\alpha_{k}^{2}(2)\leq\frac{1}{8},\quad \forall k\geq k_{1}. $$
(3.15)
Then, by (3.14), we have
$$ \varphi_{\lambda}(u,v)\geq\frac{3}{8}\bigl\| (u,v)\bigr\| ^{2}-2C \bigl(\alpha _{k}^{p}(2)\bigl\| (u,v)\bigr\| ^{p}+ \alpha_{k}^{q}(2)\bigl\| (u,v)\bigr\| ^{q} \bigr). $$
(3.16)
Let
$$ \rho_{k}=\max\bigl\{ \bigl(16C\alpha_{k}^{p}(2) \bigr)^{\frac{1}{2-p}}, \bigl(16C\alpha _{k}^{q}(2) \bigr)^{\frac{1}{2-q}}\bigr\} . $$
(3.17)
Obviously, \(\rho_{k}\rightarrow0\) as \(k\rightarrow\infty\) since \(p, q\in(1,2)\). By (3.16) and (3.17) direct computation shows that
$$a_{k}(\lambda) :=\inf_{(u,v)\in Z_{k}\times Z_{k}, \|(u,v)\|= \rho_{k}}\varphi_{\lambda}(u,v) \geq\frac{\rho_{k}^{2}}{8}>0,\quad \forall k\geq k_{1}. $$
Moreover, by (3.14), for any \((u,v)\in Z_{k}\times Z_{k}\) with \(\|(u,v)\|= \rho_{k}\), we have
$$\varphi_{\lambda}(u,v)\geq-2C\bigl(\alpha_{k}^{2}(2) \bigl\| (u,v)\bigr\| ^{2}+\alpha _{k}^{p}(2)\bigl\| (u,v) \bigr\| ^{p}+\alpha_{k}^{q}(2)\bigl\| (u,v)\bigr\| ^{q} \bigr). $$
Therefore,
$$0\geq\inf_{(u,v)\in Z_{k}\times Z_{k}, \|(u,v)\|\leq\rho _{k}} \varphi_{\lambda}(u,v)\geq-2C\bigl( \alpha_{k}^{2}(2)\bigl\| (u,v)\bigr\| ^{2}+\alpha _{k}^{p}(2)\bigl\| (u,v)\bigr\| ^{p}+\alpha_{k}^{q}(2) \bigl\| (u,v)\bigr\| ^{q}\bigr). $$
Since \(\alpha_{k}(2)\rightarrow0\) as \(k\rightarrow\infty\), we have
$$d_{k}(\lambda):=\inf_{(u,v)\in Z_{k}\times Z_{k}, \|(u,v)\|\leq \rho_{k}}\varphi_{\lambda}(u,v) \rightarrow0 \quad\mbox{as } k\rightarrow \infty \mbox{ uniformly for } \lambda\in[1,2]. $$
The proof is completed. □

Lemma 3.3

Suppose that (H1)-(H4) hold. Then for the positive integer \(k_{1}\) and the sequence \(\{\rho_{k}\}\) obtained in Lemma  3.2, for all \(k\geq k_{1}\), there exist \(0< r_{k}<\rho_{k}\) such that
$$b_{k}(\lambda):=\max_{(u,v)\in Y_{k}\times Y_{k}, \|(u,v)\|=r_{k}}\varphi_{\lambda}(u,v)< 0\quad \textit{for all } \lambda\in [1,2], \forall k\geq k_{1}, $$
where \(Y_{k}=\bigoplus_{j=1}^{k}X_{j}=\operatorname{span}\{e_{1},\ldots e_{k}\}\) for all \(k\in N\).

Proof

Note that \(Y_{k}\times Y_{k}\) is a finite-dimensional subspace of E. Then by (3.3) there exists a constant \(\varepsilon_{k}\) such that
$$ \operatorname{meas} \bigl(\Omega_{(u,v)}^{k}\bigr)\geq \varepsilon_{k}, \quad\forall(u,v)\in Y_{k}\times Y_{k}\setminus\bigl\{ (0,0)\bigr\} , $$
(3.18)
where
$$\Omega_{(u,v)}^{k}=\operatorname{meas} \bigl\{ x\in \mathbb{R}^{N}: H(x)\bigl|(u,v)\bigr|^{\mu }\geq\varepsilon_{k} \bigl\| (u,v)\bigr\| ^{\mu}\bigr\} ,\quad \forall(u,v)\in Y_{k}\times Y_{k}\setminus\bigl\{ (0,0)\bigr\} . $$
Combining (3.2), (H3), (H4), and (3.18), for any \(k\in N\) and \(\lambda\in[1,2]\), we have
$$ \begin{aligned}[b] \varphi_{\lambda}(u,v) &=\frac{1}{2}\bigl\| (u,v) \bigr\| ^{2}-\lambda \int_{\mathbb{R}^{N}}H(x)F(x, u, v)\,dx \\ &\leq\frac{1}{2}\bigl\| (u,v)\bigr\| ^{2}-c_{1} \int_{\mathbb{R}^{N}}H(x)\bigl|(u,v)\bigr|^{\mu }\,dx \\ &\leq\frac{1}{2}\bigl\| (u,v)\bigr\| ^{2}-c_{1} \int_{\Omega _{(u,v)}^{k}}H(x)\bigl|(u,v)\bigr|^{\mu}\,dx \\ &\leq\frac{1}{2}\bigl\| (u,v)\bigr\| ^{2}-c_{1} \varepsilon_{k}\bigl\| (u,v)\bigr\| ^{\mu} \operatorname{meas} \bigl( \Omega_{(u,v)}^{k}\bigr) \\ &\leq\frac{1}{2}\bigl\| (u,v)\bigr\| ^{2}-c_{1} \varepsilon_{k}^{2}\bigl\| (u,v)\bigr\| ^{\mu}. \end{aligned} $$
(3.19)
For \(\|(u,v)\|=r_{k}<\rho_{k}\) small enough, we have
$$b_{k}(\lambda):=\max_{(u,v)\in Y_{k}\times Y_{k}, \|(u,v)\|=r_{k}}\varphi_{\lambda}(u,v)< 0 \quad\mbox{for all } \lambda\in [1,2], \forall k\geq k_{1}, $$
since \(\mu\in(1,2)\). The proof is completed. □

Now we give the proof of Theorem 1.1.

Proof

Obviously, condition (C1) in Theorem 2.1 holds. By Lemmas 3.1-3.3 conditions (C2) and (C3) in Theorem 2.1 are also satisfied. Furthermore, by Theorem 2.1, there exist \(\lambda_{n}\rightarrow1\) and \((u(\lambda_{n}),v(\lambda _{n}))\in Y_{n}\times Y_{n}\) such that
$$ \begin{aligned} &\varphi_{\lambda_{n}}'|_{Y_{n}\times Y_{n}}\bigl(u( \lambda_{n}),v(\lambda_{n})\bigr)=0, \\ &\varphi_{\lambda_{n}} \bigl(u(\lambda_{n}),v(\lambda_{n})\bigr) \rightarrow c_{k}\in\bigl[d_{k}(2),b_{k}(1)\bigr] \quad\mbox{as } n\rightarrow\infty . \end{aligned} $$
(3.20)

For simplicity, in what follows, we always set \((u_{n},v_{n})=(u(\lambda_{n}),v(\lambda_{n}))\) for all \(n\in\mathbb {N}\).

Now we claim that the sequence \(\{(u_{n},v_{n})\}\) obtained in (3.19) is bounded in E. Indeed, by (H2), (H4), (3.2), (3.20), and the Hölder inequality we have
$$ \begin{aligned}[b]\bigl\| (u_{n},v_{n})\bigr\| ^{2} &\leq2 \varphi_{\lambda_{n}}\bigl((u_{n},v_{n})\bigr)+2 \lambda_{n} \int_{\mathbb {R}^{N}}H(x)F(x,u_{n},v_{n})\,dx \\ &\leq C_{0}+C\bigl(\bigl\| (u_{n},v_{n}) \bigr\| ^{2}+\bigl\| (u,v)\bigr\| _{p}^{p}+\bigl\| (u,v) \bigr\| _{q}^{q}\bigr) \end{aligned} $$
(3.21)
for some \(C_{0}>0\). Since \(p, q\in(1,2)\), (3.21) implies that \(\{(u_{n},v_{n})\}\) is bounded in E.
Finally, we show that \(\{(u_{n},v_{n})\}\) possesses a strong convergent sequence in E. Indeed, since \(\{(u_{n},v_{n})\}\) is bounded, there exists \((u_{0},v_{0})\in E\) such that
$$\begin{aligned}& (u_{n},v_{n})\rightharpoonup(u_{0},v_{0}) \quad\mbox{in } E,\\& (u_{n},v_{n})\rightarrow(u_{0},v_{0}) \quad\mbox{in } L^{p}\bigl(\mathbb {R}^{N}\bigr)\times L^{p}\bigl(\mathbb {R}^{N}\bigr), p\in\bigl[2,2^{*}\bigr),\\& (u_{n},v_{n})\rightarrow(u_{0},v_{0}) \quad\mbox{a.e. on } \mathbb {R}^{N}. \end{aligned}$$
By (2.2) we easily get
$$ \begin{aligned}[b] \bigl\| (u_{n},v_{n})-(u_{0},v_{0}) \bigr\| ^{2} ={}&\bigl\langle \varphi_{\lambda_{n}}'(u_{n},v_{n})- \varphi _{1}'(u_{0},v_{0}),(u_{n},v_{n})-(u_{0},v_{0}) \bigr\rangle \\ &{} + \int_{\mathbb {R}^{N}}H(x) \bigl(\lambda _{n}F_{u}(x,u_{n},v_{n})-F_{u}(x,u_{0},v_{0}) \bigr) (u_{n}-u_{0})\,dx \\ &{} + \int_{\mathbb {R}^{N}}H(x) \bigl(\lambda _{n}F_{v}(x,u_{n},v_{n})-F_{v}(x,u_{0},v_{0}) \bigr) (v_{n}-v_{0})\,dx. \end{aligned} $$
(3.22)
Clearly,
$$ \bigl\langle \varphi_{\lambda_{n}}'(u_{n},v_{n})- \varphi _{1}'(u_{0},v_{0}),(u_{n},v_{n})-(u_{0},v_{0}) \bigr\rangle \rightarrow0. $$
(3.23)
Denote
$$\begin{aligned}& M:= \int_{\mathbb {R}^{N}}H(x) \bigl(\lambda _{n}F_{u}(x,u_{n},v_{n})-F_{u}(x,u_{0},v_{0}) \bigr) (u_{n}-u_{0})\,dx,\\& N:= \int_{\mathbb {R}^{N}}H(x) \bigl(\lambda _{n}F_{v}(x,u_{n},v_{n})-F_{v}(x,u_{0},v_{0}) \bigr) (v_{n}-v_{0})\,dx. \end{aligned}$$
Then by (H2), (H4), and the Hölder and Minkowski inequalities we have
$$\begin{aligned} M \leq{}& C\|u_{n}-u_{0}\|_{2} \biggl( \int_{\mathbb {R}^{N}}H^{2}(x) \bigl(2\bigl|(u_{n},v_{n})\bigr| +2\bigl|(u_{n},v_{n})\bigr|^{p-1}+2\bigl|(u_{n},v_{n})\bigr|^{q-1} \\ &{} +\bigl|(u_{0},v_{0})\bigr|+\bigl|(u_{0},v_{0})\bigr|^{p-1}+\bigl|(u_{0},v_{0})\bigr|^{q-1} \bigr)^{2}\,dx \biggr)^{\frac{1}{2}} \\ \leq{}& C\|u_{n}-u_{0}\|_{2} \biggl( \int_{\mathbb {R}^{N}}H^{2}(x) \bigl(2\bigl|(u_{n},v_{n})\bigr|+\bigl|(u_{0},v_{0})\bigr| \bigr)^{2}+H^{2}(x) \bigl(2\bigl|(u_{n},v_{n})\bigr|^{p-1} \\ &{} +\bigl|(u_{0},v_{0})\bigr|^{p-1}\bigr)^{2}+H^{2}(x) \bigl(2\bigl|(u_{n},v_{n})\bigr|^{q-1}+\bigl|(u_{0},v_{0})\bigr|^{q-1} \bigr)^{2}\,dx \biggr)^{\frac {1}{2}} \\ \leq{}& C\|u_{n}-u_{0}\|_{2} \biggl( \int_{\mathbb {R}^{N}}H^{2}(x) \bigl(4\bigl|(u_{n},v_{n})\bigr|^{2} +\bigl|(u_{0},v_{0})\bigr|^{2} \bigr)+H^{2}(x) \bigl(4\bigl|(u_{n},v_{n})\bigr|^{2p-2} \\ &{} +\bigl|(u_{0},v_{0})\bigr|^{2p-2}\bigr)+H^{2}(x) \bigl(4\bigl|(u_{n},v_{n})\bigr|^{2q-2}+\bigl|(u_{0},v_{0})\bigr|^{2q-2} \bigr)\,dx \biggr)^{\frac{1}{2}} \\ \leq{}& C\|u_{n}-u_{0}\|_{2} \biggl[ \biggl( \int_{\mathbb {R}^{N}}4H^{2}(x)\bigl|(u_{n},v_{n})\bigr|^{2} \,dx \biggr)^{\frac{1}{2}}+ \biggl( \int _{\mathbb {R}^{N}}H^{2}(x)\bigl|(u_{0},v_{0})\bigr|^{2} \,dx \biggr)^{\frac{1}{2}} \\ &{} + \biggl( \int_{\mathbb {R}^{N}}4H^{2}(x)\bigl|(u_{n},v_{n})\bigr|^{2p-2} \biggr)^{\frac{1}{2}}+ \biggl( \int_{\mathbb {R}^{N}}H^{2}(x)\bigl|(u_{0},v_{0})\bigr|^{2p-2} \,dx \biggr)^{\frac{1}{2}} \\ &{} + \biggl( \int_{\mathbb {R}^{N}}4H^{2}(x)\bigl|(u_{n},v_{n})\bigr|^{2q-2} \,dx \biggr)^{\frac{1}{2}}\,dx+ \biggl( \int_{\mathbb {R}^{N}}H^{2}(x)\bigl|(u_{0},v_{0})\bigr|^{2q-2} \,dx \biggr)^{\frac{1}{2}} \biggr] \\ \leq{}& C\|u_{n}-u_{0}\|_{2} \bigl[2\|H \|_{\infty }\bigl\| (u_{n},v_{n})\bigr\| _{2}^{2}+ \|H\|_{\infty}\bigl\| (u_{0},v_{0})\bigr\| _{2}^{2} +2\|H\|_{\frac{2}{2-p}}\bigl\| (u_{n},v_{n})\bigr\| _{2}^{p-1} \\ &{} +\|H\|_{\frac{2}{2-p}}\bigl\| (u_{0},v_{0})\bigr\| _{2}^{p-1}+2 \|H\|_{\frac {2}{2-q}}\bigl\| (u_{n},v_{n})\bigr\| _{2}^{q-1}+ \|H\|_{\frac{2}{2-q}}\bigl\| (u_{0},v_{0})\bigr\| _{2}^{q-1} \bigr] \\ \leq{}& C\|u_{n}-u_{0}\|_{2} \bigl( \bigl\| (u_{n},v_{n})\bigr\| _{2}^{2}+ \bigl\| (u_{0},v_{0})\bigr\| _{2}^{2}+ \bigl\| (u_{n},v_{n})\bigr\| _{2}^{p-1} + \bigl\| (u_{0},v_{0})\bigr\| _{2}^{p-1} \\ &{} +\bigl\| (u_{n},v_{n})\bigr\| _{2}^{q-1}+ \bigl\| (u_{0},v_{0})\bigr\| _{2}^{q-1} \bigr). \end{aligned}$$
(3.24)
Since \((u_{n},v_{n})\rightarrow(u_{0},v_{0})\) in \(L^{p}(\mathbb {R}^{N})\times L^{p}(\mathbb {R}^{N})\), for any \(p\in[2,2^{*})\), we obtain
$$ \int_{\mathbb {R}^{N}}H(x) \bigl(\lambda _{n}F_{u}(x,u_{n},v_{n})-F_{u}(x,u_{0},v_{0}) \bigr) (u_{n}-u_{0})\,dx\rightarrow 0 \quad\mbox{as } n\rightarrow \infty. $$
(3.25)
Similarly, we can also obtain
$$ \int_{\mathbb {R}^{N}}H(x) \bigl(\lambda _{n}F_{v}(x,u_{n},v_{n})-F_{v}(x,u_{0},v_{0}) \bigr) (v_{n}-v_{0})\,dx\rightarrow 0 \quad\mbox{as } n\rightarrow \infty. $$
(3.26)
Therefore, by (3.22)-(3.26) we get \(\|(u_{n},v_{n})-(u_{0},v_{0})\| \rightarrow0\) as \(n\rightarrow\infty\).

Now from the last assertion of Theorem 2.1 we know that \(I=\varphi_{1}\) has infinitely many nontrivial critical points. Therefore, system (1.1) possesses infinitely many small negative-energy solutions. The proof is completed. □

Declarations

Acknowledgements

This work is partially supported by Natural Science Foundation of China 11271372 and Mathematics and Interdisciplinary Sciences Project of CSU.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Central South University

References

  1. De Figueiredo, DG, Felmer, PL: On superquadratic elliptic systems. Trans. Am. Math. Soc. 223, 99-116 (1994) MathSciNetView ArticleMATHGoogle Scholar
  2. De Figueiredo, DG, Mitidieri, E: A maximum principle for an elliptic system and applications to semilinear problem. SIAM J. Math. Anal. 17, 836-849 (1986) MathSciNetView ArticleMATHGoogle Scholar
  3. Lazer, AC, McKenna, PJ: On steady-state solutions of a system of reaction-diffusion equations from biology. Nonlinear Anal. TMA 6, 523-530 (1982) MathSciNetView ArticleMATHGoogle Scholar
  4. Silva, EA: Existence and multiplicity of solutions for semilinear elliptic systems. NoDEA Nonlinear Differ. Equ. Appl. 1, 339-363 (1994) MathSciNetView ArticleMATHGoogle Scholar
  5. Qu, Z, Tang, C: Existence and multiplicity results for some elliptic systems at resonance. Nonlinear Anal. 71, 2660-2666 (2009) MathSciNetView ArticleMATHGoogle Scholar
  6. Duan, S, Wu, X: The existence of solutions for a class of semilinear elliptic systems. Nonlinear Anal. 73, 2842-2854 (2010) MathSciNetView ArticleMATHGoogle Scholar
  7. Pomponio, A: Asymptotically linear cooperative elliptic system: existence and multiplicity. Nonlinear Anal. 52, 989-1003 (2003) MathSciNetView ArticleMATHGoogle Scholar
  8. Costa, DG, Magalhães, CA: A unified approach to a class of strong indefinite functions. J. Differ. Equ. 125, 521-547 (1996) View ArticleMATHGoogle Scholar
  9. Costa, DG, Magalhães, CA: A variational method to subquadratic perturbations of elliptic systems. J. Differ. Equ. 111, 103-122 (1994) View ArticleMATHGoogle Scholar
  10. Cao, D, Tang, Z: Solutions with prescribed number of nodes to superlinear elliptic systems. Nonlinear Anal. 55, 702-722 (2003) MathSciNetView ArticleMATHGoogle Scholar
  11. Zhang, Z, Shi, Y, Xue, Y: Existence of entire solutions for semilinear elliptic systems under the Keller-Osserman condition. Electron. J. Differ. Equ. 2011, 39 (2011) MathSciNetView ArticleMATHGoogle Scholar
  12. Wu, X: High energy solutions of systems of Kirchhoff-type equations in \(\mathbb {R}^{N}\). J. Math. Phys. 53, 063508 (2012) MathSciNetView ArticleMATHGoogle Scholar
  13. Zhou, F, Wu, K, Wu, X: High energy solutions of systems of Kirchhoff-type equations on \(\mathbb {R}^{N}\). Comput. Math. Appl. 66, 1299-1305 (2013) MathSciNetView ArticleGoogle Scholar
  14. Li, G, Tang, X: Nehari-type state solutions for Schrödinger equations including critical exponent. Appl. Math. Lett. 37, 101-106 (2014) MathSciNetView ArticleMATHGoogle Scholar
  15. Maia, LA, Montefusco, E, Pellacci, B: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Equ. 229, 743-767 (2006) MathSciNetView ArticleMATHGoogle Scholar
  16. Tang, X: Infinitely many solutions for semilinear Schrodinger equations with sign-changing potential and nonlinearity. J. Math. Anal. Appl. 401, 407-415 (2013) MathSciNetView ArticleMATHGoogle Scholar
  17. Huang, W, Tang, X: The existence of infinitely many solutions for the nonlinear Schrödinger-Maxwell equations. Results Math. 65, 223-234 (2014) MathSciNetView ArticleMATHGoogle Scholar
  18. Sun, J, Chen, H, Yang, L: Positive solutions of asymptotically linear Schrödinger-Poisson systems with a radial potential vanishing at infinity. Nonlinear Anal. 74, 413-423 (2011) MathSciNetView ArticleMATHGoogle Scholar
  19. Liu, H, Chen, H, Yang, X: Multiple solutions for superlinear Schrödinger-Poisson systems with sign-changing potential and nonlinearity. Comput. Math. Appl. 68, 1982-1990 (2014) MathSciNetView ArticleGoogle Scholar
  20. Zhang, J, Tang, X: Infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential. J. Math. Anal. Appl. 420, 1762-1775 (2014) MathSciNetView ArticleMATHGoogle Scholar
  21. Xu, L, Chen, H: Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type via genus theory. Bound. Value Probl. 2014, 212 (2014) MathSciNetView ArticleMATHGoogle Scholar
  22. Xu, L, Chen, H: Existence of infinitely many solutions for generalized Schrödinger-Poisson system. Bound. Value Probl. 2014, 196 (2014) View ArticleMATHGoogle Scholar
  23. Qin, D, Tang, X: New conditions on solutions for periodic Schrödinger equation with spectrum zero. Taiwan. J. Math. 19(4), 977-993 (2015) MathSciNetView ArticleGoogle Scholar
  24. Zou, W, Schechter, M: Critical Point Theory and Its Applications. Springer, New York (2006) MATHGoogle Scholar
  25. Xu, L, Chen, H: Multiplicity of small negative-energy solutions for a class of nonlinear Schrödinger-Poisson systems. Appl. Math. Comput. 243, 817-824 (2014) MathSciNetMATHGoogle Scholar

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© Che and Chen 2016