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Infinitely many solutions for p-biharmonic equation with general potential and concave-convex nonlinearity in \(\mathbb{R}^{N}\)

Boundary Value Problems20162016:6

https://doi.org/10.1186/s13661-015-0510-6

©  Liu and Chen 2016

Received: 2 September 2015

Accepted: 28 December 2015

Published: 12 January 2016

Abstract

In this paper, we study the existence of multiple solutions to a class of p-biharmonic elliptic equations, \(\Delta^{2}_{p}u-\Delta_{p}u+V(x)|u|^{p-2}u =\lambda h_{1}(x)|u|^{m-2}u+h_{2}(x)|u|^{q-2}u\), \(x\in \mathbb{R}^{N}\), where \(1< m< p< q< p_{*}=\frac {pN}{N-2p}\), \(\Delta^{2}_{p} u=\Delta(|\Delta u|^{p-2}\Delta u)\) is a p-biharmonic operator and \(\Delta_{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\). The potential function \(V(x)\in C{({\mathbb{R}}^{N})}\) satisfies \(\inf_{x\in{\mathbb{R}} ^{N}}V(x)>0\). By variational methods, we obtain the existence of infinitely many solutions for a p-biharmonic elliptic equation in \({\mathbb{R}}^{N}\).

Keywords

p-biharmonic elliptic equation variational methods mountain pass lemma

1 Introduction

In this paper, we are interested in the existence of solutions to the following p-biharmonic elliptic equation:
$$ \Delta^{2}_{p}u-\Delta_{p}u+V(x)|u|^{p-2}u =f(x,u),\quad x\in\mathbb{R}^{N}, $$
(1.1)
where \(2<2p<N\), \(f(x,u)=\lambda h_{1}(x)|u|^{m-2}u+h_{2}(x)|u|^{q-2}u\), \(1< m< p< q< p_{*}=\frac{pN}{N-2p}\), \(\Delta^{2}_{p} u=\Delta(|\Delta u|^{p-2}\Delta u)\) is a p-biharmonic operator and \(\Delta _{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\). The potential function \(V(x)\in C{({\mathbb{R}}^{N})}\) satisfies \(\inf_{x\in{\mathbb{R}}^{N}}V(x)>0\).
Recently, the nonlinear biharmonic equation in an unbounded domain has been extensively investigated, we refer the reader to [1–9] and the references therein. For the whole space \({\mathbb{R}}^{N}\) case, the main difficulty of this problem is the lack of compactness for the Sobolev embedding theorem. In order to overcome this difficulty, the authors always assumed the potential \(V(x)\) has some special characteristic. For example, in [4], Yin and Wu studied the following fourth-order elliptic equation:
$$ \left \{ \textstyle\begin{array}{l} \Delta^{2} u-\Delta u+V(x)u =f(x,u), \quad x\in\mathbb{R}^{N}, \\ u(x)\in H^{2}({{\mathbb{R}}^{N}}), \end{array}\displaystyle \right . $$
(1.2)
where the potential \(V(x)\) satisfied
(V0):: 

\(V\in C({\mathbb{R}}^{N})\) satisfies \(\inf_{x\in{\mathbb {R}}^{N}}V(x)>0\), and for each \(M>0\), \(\operatorname{meas}\{x\in \mathbb{R}^{N}\leq M\}<+\infty\).

This assumption guarantees that the embedding \(H^{2}\hookrightarrow L^{s}({\mathbb{R}}^{N})\) is compact for each \(s\in[2,\frac{2N}{N-4})\) and obeys the coercivity condition: \(V(x)\to\infty\) as \(|x|\to\infty\). Hence, under various sets of assumptions on the nonlinear term \(f(x, t)\) (subcriticality, superquadraticity, etc.), the authors proved the existence of infinitely many solutions to problem (1.2) by using the variational techniques in a standard way. In [2], Liu et al. considered the following fourth-order elliptic equation:
$$ \left \{ \textstyle\begin{array}{l} \Delta^{2} u-\Delta u+\lambda V(x)u =f(x,u),\quad x\in\mathbb{R}^{N}, \\ u(x)\in H^{2}({{\mathbb{R}}^{N}}), \end{array}\displaystyle \right . $$
(1.3)
where the potential \(V(x)\) satisfied a weaker condition than (V0), that is,
(V1):: 

\(V\in C({\mathbb{R}}^{N})\) satisfies \(\inf_{x\in{\mathbb {R}}^{N}}V(x)>0\), there exists some \(M>0\), \(\operatorname{meas}\{x\in \mathbb{R}^{N}\leq M\}<+\infty\).

Under the assumption (V1), the compactness of the embedding is lost and this renders variational techniques more delicate. With the aid of the parameter \({\lambda}>0\), they proved that the variational functional satisfies \((\mathit{PS})\) condition, and then they showed the existence and multiplicity results of problem (1.3). A natural question is whether the existence results still holds if we assume a more general potential \(V(x)\) than (V0), (V1), namely,
(V):: 

\(V(x)\in C({\mathbb{R}}^{N})\) satisfies \(\inf_{x\in {\mathbb{R}}^{N}}V(x)>0\).

In the present paper, we will answer this interesting question. We consider the existence of solutions to the p-biharmonic problem (1.1) with a more general potential \(V(x)\). To prove that the \((\mathit{PS})\) sequence weakly converges to a critical point of the corresponding functional, we adapt ideas developed by [10–12] and then by variational methods, we establish the existence of infinitely many high-energy solutions to problem (1.1) with a concave-convex nonlinearity, i.e., \(f(x,u)=\lambda h_{1}(x)|u|^{m-2}u+h_{2}(x)|u|^{q-2}u\), \(1< m< p< q< p_{*}=\frac{pN}{N-2p}\). To the best of our knowledge, little has been done for p-biharmonic problems with this type of nonlinearity. Here, we give our assumptions on the weight functions \(h_{1}(x)\) and \(h_{2}(x)\):
(H1): 

\(h_{1}\in L^{\sigma}({\mathbb{R}}^{N})\) with \(\sigma=\frac {p}{p-m}\);

(H2): 

\(h_{2}(x)\geq0\) (≢0), \(h_{2}(x)\in L^{\infty}({\mathbb{R}}^{N})\).

The main result in this paper is as follows.

Theorem 1.1

Let \(2<2p<N\), \(1< m< p< q< p_{*}=\frac{pN}{N-2p}\). Assume (V), (H1), and (H2) hold. Then there exists \(\lambda_{0}>0\) such that for all \(\lambda\in [0,\lambda_{0}]\), problem (1.1) admits infinitely many high-energy solutions in \({\mathbb{R}}^{N}\).

This paper is organized as follows. In Section 2, we build the variational framework for problem (1.1) and establish a series of lemmas, which will be used in the proof of Theorem 1.1. In Section 3, we prove Theorem 1.1 by the mountain pass theorem [13].

2 Preliminaries

In order to apply the variational setting, we assume the solutions of (1.1) belong to the following subspace of \(\mathcal {D}^{2,p}({\mathbb{R}}^{N})\):
$$ E=\biggl\{ u\in\mathcal{D}^{2,p}\bigl({ \mathbb{R}}^{N}\bigr)\Big| \int_{{\mathbb {R}}^{N}}|\Delta u|^{p}+|\nabla u|^{p}+V(x)|u|^{p} \,dx< \infty\biggr\} $$
(2.1)
endowed with the norm
$$ \|u\|_{E}=\biggl( \int_{{\mathbb{R}}^{N}}\bigl(|\Delta u|^{p}+|\nabla u|^{p}+V(x)|u|^{p}\bigr)\, dx\biggr)^{1/p}, $$
(2.2)
where \(\mathcal{D}^{2,p}({\mathbb{R}}^{N})=\{u\in L^{p_{*}}({\mathbb {R}}^{N}) |\Delta u\in L^{p}({\mathbb{R}}^{N})\}\), \(\|\cdot\|_{s}\) means the norm in \(L^{s}({\mathbb{R}}^{N})\).
We denote by \(S_{*}\) the Sobolev constant, that is,
$$ S_{*}=\inf_{u\in\mathcal{D}^{2,p}\setminus\{0\}}\frac{\int _{{\mathbb{R}}^{N}}|\Delta u|^{p}\,dx}{(\int_{{\mathbb {R}}^{N}}|u|^{p_{*}}\, dx)^{p/p_{*}}} $$
(2.3)
and
$$ S_{*}\biggl( \int_{{\mathbb{R}}^{N}}|u|^{p_{*}}\, dx\biggr)^{p/p_{*}}\leq \int_{{\mathbb {R}}^{N}}|\Delta u|^{p}\,dx,\quad \forall u\in \mathcal{D}^{2,p}\bigl({\mathbb{R}}^{N}\bigr), $$
(2.4)
where \(S_{*}\) is obtained by a positive and radially symmetric function; see for instance [14].

Definition 2.1

A function \(u\in E\) is said to be a weak solution of (1.1) if, for any \(\varphi\in E\), we have
$$\begin{aligned}& \int_{{\mathbb{R}}^{N}}\bigl(|\Delta u|^{p-2}\Delta u\Delta\varphi+| \nabla u|^{p-2}\nabla u\varphi+V|u|^{p-2}u\varphi\bigr)\, dx \\& \quad = \int_{{\mathbb{R}} ^{N}}\bigl(\lambda h_{1}(x)|u|^{m-2}u+h_{2}(x)|u|^{q-2}u \bigr)\varphi\, dx. \end{aligned}$$
(2.5)
Let \(J(u):E\to{\mathbb{R}}\) be the energy functional associated with problem (1.1) defined by
$$ J(u)=\frac{1}{p}\|u\|^{p}_{E}- \frac{\lambda}{m} \int_{{\mathbb{R}} ^{N}}h_{1}|u|^{m}\,dx-\frac{1}{q} \int_{{\mathbb{R}}^{N}}h_{2}|u|^{q}\,dx. $$
(2.6)
From the embedding inequality (2.4) and the assumptions in Theorem 1.1, we see the functional \(J\in C^{1}(E,{\mathbb{R}})\) and its Gateaux derivative is given by
$$\begin{aligned} J'(u)\varphi =& \int_{{\mathbb{R}}^{N}}\bigl(|\Delta u|^{p-2}\Delta u\Delta \varphi +|\nabla u|^{p-2}\nabla u\varphi+V(x)|u|^{p-2}u\varphi\bigr)\, dx \\ &{}- \int_{{\mathbb{R}}^{N}}\bigl(\lambda h_{1}(x)|u|^{m-2}u+h_{2}(x)|u|^{q-2}u \bigr)\varphi\, dx. \end{aligned}$$
(2.7)
To prove the existence of infinitely many solutions to problem (1.1), we need to prove that the functional J defined by (2.6) satisfies the \((\mathit{PS})\) condition. Recall that a sequence \(\{u_{n}\}\) in E is called a \((\mathit{PS})_{c}\) sequence of J if
$$ J(u_{n})\to c, \qquad J'(u_{n}) \to0 \quad \mbox{in }E^{*}\mbox{ as }n\to\infty. $$
(2.8)
The functional J satisfies the \((\mathit{PS})\) condition if any \((\mathit{PS})_{c}\) sequence possesses a convergent subsequence in E.

Lemma 2.1

Assume (V), (H1), and (H2) hold. If \(\{u_{n}\}\subset E\) is a \((\mathit{PS})_{c}\) sequence of J, then \(\{u_{n}\}\) is bounded in E.

Proof

It follows from Hölder’s inequality that
$$\begin{aligned} \int_{{\mathbb{R}}^{N}}|h_{1}||u_{n}|^{m}\,dx \leq&V_{0}^{-\frac{m}{p}}\biggl( \int _{{\mathbb{R}} ^{N}}|h_{1}|^{\sigma}\, dx \biggr)^{\frac{1}{\sigma}}\biggl( \int_{{\mathbb{R}} ^{N}}V|u_{n}|^{p}\,dx \biggr)^{\frac{m}{p}} \\ \leq&a_{1}\|u_{n}\|^{m}_{E}, \end{aligned}$$
(2.9)
where \(a_{1}=V_{0}^{-{m}/{p}}\|h_{1}\|_{\sigma}\). Choose \(t\in(0,1)\) such that \(q=pt+(1-t)p_{*}\), then
$$\begin{aligned} \int_{{\mathbb{R}}^{N}}h_{2}|u_{n}|^{q}\,dx \leq&\biggl( \int_{{\mathbb {R}}^{N}}V|u_{n}|^{p}\,dx \biggr)^{t}\biggl( \int_{{\mathbb{R}} ^{N}}|u_{n}|^{p_{*}}h_{2}^{-{\frac{1}{1-t}}}V^{-\frac{t}{1-t}} \, dx\biggr)^{1-t} \\ \leq& a_{2}\biggl( \int_{{\mathbb{R}}^{N}}V|u_{n}|^{p}\,dx \biggr)^{t}\|\Delta u_{n}\| _{p}^{{(1-t)}p_{*}}\leq a_{2}\|u_{n}\|^{q}_{E}, \end{aligned}$$
(2.10)
where \(a_{2}=S_{*}^{-{p_{*}(q-p)}/{p(p_{*}-p)}}V_{0}^{-t}\|h_{2}\|_{\infty}\). Thus,
$$\begin{aligned} c+1+\|u_{n}\|_{E} \geq& J(u_{n})-q^{-1}J'(u_{n})u_{n} \\ \geq&\biggl(\frac{1}{p}-\frac{1}{q}\biggr)\|u\|_{E}^{p}- \lambda\biggl(\frac{1}{m}-\frac {1}{q}\biggr) \int_{{\mathbb{R}}^{N}}|h_{1}||u|^{m}\,dx \\ \geq&\biggl(\frac{1}{p}-\frac{1}{q}\biggr)\|u_{n} \|^{p}_{E}-\lambda\biggl(\frac {1}{m}-\frac{1}{q} \biggr)a_{1}\|u_{n}\|^{m}_{E}. \end{aligned}$$
(2.11)
Since \(1< m< p< q\), we conclude that \(\|u\|_{E}\) is bounded and the proof is complete. □
In the following, we shall show that \(\{u_{n}\}\) has a convergent subsequence in E. Since the sequence \(\{u_{n}\}\) given by (2.8) is a bounded sequence in E, there exist a subsequence of \(\{u_{n}\}\) (still denoted by \(\{u_{n}\}\)) and \(v\in E\) such that \(\|u_{n}\|_{E}\leq M\), \(\|v\|_{E}\leq M\), and
$$\begin{aligned}& u_{n}\rightharpoonup v \quad \text{weakly in }E, \\& u_{n}\to v \quad \text{in } L^{s}_{\mathrm{loc}}\bigl({ \mathbb{R}}^{N}\bigr), 1< s< p_{*}, \\& u_{n}(x)\to v(x)\quad \text{a.e. in } {\mathbb{R}}^{N}. \end{aligned}$$
(2.12)

Lemma 2.2

Assume (V), (H1), and (H 2) hold. If the sequence \(\{u_{n}\}\) is bounded in E satisfying (2.12), then
  1. (i)

    \(\lim_{n\to\infty}\int_{{\mathbb{R}} ^{N}}h_{1}(x)|u_{n}|^{m}\,dx=\int_{{\mathbb{R}}^{N}}h_{1}(x)|v|^{m}\,dx\), \(\lim_{n\to \infty}\int_{{\mathbb{R}}^{N}}h_{1}(x)|u_{n}-v|^{m}\,dx=0\);

     
  2. (ii)

    \(\lim_{n\to\infty}\int_{{\mathbb{R}} ^{N}}h_{2}(x)|u_{n}|^{q}\,dx=\int_{{\mathbb{R}}^{N}}h_{2}(x)|v|^{q}\,dx\), \(\lim_{n\to \infty}\int_{{\mathbb{R}}^{N}}h_{2}(x)|u_{n}-v|^{q}\,dx=0\).

     

Proof

(i) In fact, from \(h\in L^{\sigma}({\mathbb{R}}^{N})\) and (2.12), we obtain, for any \(r>0\),
$$ \int_{B_{r}}h_{1}(x)|u_{n}|^{m}\,dx \to \int_{B_{r}}h_{1}(x)|v|^{m}\,dx \quad \mbox{as }n\to \infty, $$
(2.13)
where and in the sequel \(B_{r}=\{x\in{\mathbb{R}}^{N}:|x|< r\}\), \(B^{c}_{r}={\mathbb{R}}^{N}\setminus \overline{B}_{r}\). On the other hand, we see from the Hölder inequality that
$$\begin{aligned} \begin{aligned}[b] \int_{B^{c}_{r}}|h_{1}||u_{n}|^{m}\,dx& \leq V_{0}^{-\frac{m}{p}}\biggl( \int _{B^{c}_{r}}|h_{1}|^{\sigma}\, dx \biggr)^{\frac{1}{\sigma}}\biggl( \int _{B^{c}_{r}}V|u_{n}|^{p}\,dx \biggr)^{\frac{m}{p}} \\ &\leq V_{0}^{-\frac{m}{p}}\|h_{1}\|_{L^{\sigma}(B^{c}_{r})} \|u_{n}\|^{m}_{E}\leq V_{0}^{-\frac{m}{p}}M^{m} \|h_{1}\|_{L^{\sigma}(B^{c}_{r})}\to0 \end{aligned} \end{aligned}$$
(2.14)
as \(r\to\infty\). By Fatou’s lemma, we see that, as \(n\to\infty\),
$$ \int_{B^{c}_{r}}|h_{1}||v|^{m}\,dx\leq\liminf _{n\to\infty} \int _{B^{c}_{r}}|h_{1}||u_{n}|^{m}\,dx \leq V_{0}^{-\frac{m}{p}}M^{m}\|h_{1} \|_{L^{\sigma}(B^{c}_{r})}\to0. $$
(2.15)
Then, the application of (2.13)-(2.15) gives the first limit of (i). Furthermore, by the Brezis-Lieb lemma in [15], we have the second limit of (i).

(ii) To prove the conclusion (ii), we follow the argument used in [10–12]. Here, we give a detailed proof for the reader’s convenience.

Since \(p< p< p_{*}\), it easy to see that, for any small \(\varepsilon>0\), there exist \(S_{0}>s_{0}>0\) such that \(|s|^{q}<\varepsilon|s|^{p}\) if \(|s|\leq s_{0}\) and \(|s|^{q}\leq\varepsilon|s|^{p_{*}}\), if \(|s|\geq S_{0}\). This shows that
$$ |s|^{q}\leq\varepsilon\bigl(|s|^{p}+|s|^{p_{*}} \bigr)+\chi _{[s_{0},S_{0}]}\bigl(\vert s\vert \bigr)|s|^{q}, \quad \forall s\in{\mathbb{R}}. $$
(2.16)
Denote \(A_{n}=\{x\in{\mathbb{R}}^{N};s_{0}\leq|u_{n}(x)|\leq S_{0}\}\). It follows from (2.16), (2.4), and (2.12) that
$$\begin{aligned} \int_{B^{c}_{r}}|h_{2}||u_{n}|^{q}\,dx \leq&\|h_{2}\|_{\infty}\int_{B^{c}_{r}} \bigl(\varepsilon\bigl(|u_{n}|^{p}+|u_{n}|^{p_{*}} \bigr)+\chi_{[s_{0},S_{0}]}\bigl(\vert u_{n}\vert \bigr)|u_{n}|^{q} \bigr)\, dx \\ \leq&\varepsilon\|h_{2}\|_{\infty}\biggl( \int_{{\mathbb{R}} ^{N}}V_{0}^{-1}V(x)|u_{n}|^{p} \,dx+S^{-\frac{p_{*}}{p}}\|\Delta u\|^{p_{*}}_{p}\biggr) \\ &{} +S_{0}^{q}\|h_{2}\|_{\infty}\operatorname{meas}\bigl(A_{n}\cap B^{c}_{r}\bigr) \\ \leq& M_{1}\varepsilon+S_{0}^{q} \|h_{2}\|_{\infty}\operatorname{meas}\bigl(A_{n}\cap B^{c}_{r}\bigr) \end{aligned}$$
(2.17)
with some constant \(M_{1}>0\), and
$$ |s_{0}|^{p_{*}}|A_{n}|\leq \int_{{\mathbb{R}}^{N}}|u_{n}|^{p_{*}}\, dx\leq M_{1},\quad \forall n\in {\mathbb{N}}, $$
(2.18)
where \(|A_{n}|=\operatorname{meas}(A_{n})\). Equation (2.18) implies that \(\sup_{n\in{\mathbb{N}}}|A_{n}|\leq M_{1}|s_{0}|^{-p_{*}}<\infty\), so it is easy to see that
$$ \lim_{r\to\infty}\operatorname{meas} \bigl(A_{n}\cap B_{r}^{c}\bigr)=0,\quad \mbox{for all } n\in{\mathbb{N}}. $$
(2.19)
In the following, we show that \(\lim_{r\to\infty }\operatorname{meas}(A_{n}\cap B_{r}^{c})=0\) uniformly in \(n\in{\mathbb{N}}\).
In fact, it follows from (2.12) that \(v\in L^{p}({\mathbb{R}}^{N})\) and \(u_{n}(x)\to v(x)\) a.e. \({\mathbb{R}}^{N}\). Therefore, for any small \(\varepsilon >0\), there exists \(r_{0}>1\) such that \(r\geq r_{0}\),
$$\int_{B_{r}^{c}}|v|^{p}\,dx\leq\varepsilon. $$
For this ε, we choose \(t_{1}=r_{0}\), \(t_{j}\uparrow\infty\) such that \(D_{j}=B^{c}_{t_{j}}\setminus\overline{B}^{c}_{t_{j+1}}\), \(B^{c}_{r_{0}}=\bigcup^{\infty}_{j=1}D_{j}\) and
$$\int_{D_{j}}|v|^{p}\, dx\leq\frac{\varepsilon}{2^{j}},\quad \forall j\in{\mathbb{N}}. $$
Obviously, for every fixed \(j\in N\), \(D_{j}\) is a bounded domain and \(D_{j}\cap D_{i}=\emptyset\) (\(j\neq i\)). Furthermore, \(s_{0}\leq|u_{n}|\leq S_{0}\) in \(D_{j}\cap A_{n}\). By Fatou’s lemma, we have, for every \(j\in {\mathbb{N}}\),
$$\limsup_{n\to\infty} \int_{D_{j}\cap A_{n}}|u_{n}|^{p}\,dx\leq \int _{D_{j}}\limsup_{n\to\infty}|u_{n}|^{p} \,dx\leq \int _{D_{j}}|v|^{p}\,dx\leq\frac{\varepsilon}{2^{j}}. $$
Then, for \(s_{1}=2^{1-q}s_{0}^{q}\), we obtain
$$\begin{aligned} s_{1}\limsup_{n\to\infty}\bigl\vert A_{n}\cap B_{r_{0}}^{c}\bigr\vert \leq&\limsup _{n\to\infty} \int_{B_{r_{0}}^{c}\cap A_{n}}|u_{n}|^{p}\,dx \\ =&\limsup _{n\to\infty}\sum^{\infty}_{j=1} \int_{D_{j}\cap A_{n}}|u_{n}|^{p}\,dx \\ \leq&\sum^{\infty}_{j=1}\limsup _{n\to\infty} \int _{D_{j}\cap A_{n}}|u_{n}|^{p}\,dx \\ \leq&\sum ^{\infty}_{j=1} \int _{D_{j}}|v|^{p}\,dx\leq\sum ^{\infty}_{j=1}\frac{\varepsilon }{2^{j}}=\varepsilon. \end{aligned}$$
(2.20)
Notice that, for any \(r\geq r_{0}\) and \(n\in{\mathbb{N}}\), we have \((A_{n}\cap B^{c}_{r})\subset(A_{n}\cap B_{r_{0}}^{c})\). Therefore, the application of (2.19) and (2.20) yields \(\lim_{r\to\infty}|A_{n}\cap B_{r}^{c}|=0\) uniformly in \(n\in{\mathbb{N}}\). Thus, for any \(\varepsilon>0\), there exists \(r_{0}\geq1\) such that \(\operatorname{meas}(A_{n}\cap B^{c}_{r})<\frac{\varepsilon}{S_{0}^{q}\|h_{2}\| _{\infty}}\), for \(r\geq r_{0}\). Then it follows from (2.17) that
$$ \int_{B^{c}_{r}}h_{2}|u_{n}|^{q}\,dx \leq\max\{M_{1},1\}\varepsilon,\quad \forall n\in {\mathbb{N}}, r\geq r_{0} $$
(2.21)
and
$$ \int_{B^{c}_{r}}h_{2}|v|^{q}\,dx\leq\liminf _{n\to\infty} \int _{B^{c}_{r}}h_{2}|u_{n}|^{q}\,dx \leq\max\{M_{1},1\}\varepsilon, \quad r\geq r_{0}. $$
(2.22)
Moreover, we derive from (2.12) that
$$ \int_{B_{r}}h_{2}(x)|u_{n}|^{q}\,dx \to \int_{B_{r}}h_{2}(x)|v|^{q}\,dx. $$
(2.23)
Therefore, using (2.21) and (2.22), and the application of Brezis-Lieb lemma in [15] we conclude the second limit of (ii). Then the proof is complete. □

Lemma 2.3

Let \(\{u_{n}\}\) be a \((\mathit{PS})_{c}\) sequence satisfying (2.12), then \(u_{n}\to v\) in E, that is, the functional J satisfies the \((\mathit{PS})\) condition.

Proof

Denote
$$\begin{aligned} P_{n} =&J'(u_{n}) (u_{n}-v) \\ =& \int_{{\mathbb{R}}^{N}} \bigl(|\Delta u_{n}|^{p-2}\Delta u_{n} \Delta (u_{n}-v)+|\nabla u_{n}|^{p-2} \nabla u_{n}\nabla(u_{n}-u) \\ &{}+V(x)|u_{n}|^{p-2}u_{n}(u_{n}-v) \bigr)\, dx \\ &{} - \int_{{\mathbb{R}}^{N}}\bigl(\lambda h_{1}(x)|u_{n}|^{m-2}u_{n}+h_{2}(x)|u_{n}|^{q-2}u_{n} \bigr) (u_{n}-v) \, dx. \end{aligned}$$
Then the fact \(J'(u_{n})\to0\) in \(E^{*}\) shows that \(P_{n}\to0\) as \(n\to \infty\). Moreover, the fact \(u_{n}\rightharpoonup v\) in E implies \(Q_{n}\to0\), where
$$ Q_{n}= \int_{{\mathbb{R}}^{N}}\bigl(|\Delta v|^{p-2}\Delta v \Delta (u_{n}-v)+|\nabla u|^{p-2}\nabla u\nabla(u_{n}-u)+V(x)|v|^{p-2}v(u_{n}-v) \bigr)\, dx. $$
It follows from the Hölder inequality and the limit (i) in Lemma 2.2 that
$$\begin{aligned} \int_{{\mathbb{R}}^{N}}\bigl\vert h_{1}(x)\bigr\vert \vert u_{n}\vert ^{m-1}\vert u_{n}-v\vert \, dx \leq&\biggl( \int _{{\mathbb{R}} ^{N}}\bigl\vert h_{1}(x)\bigr\vert \vert u_{n}-v\vert ^{m}\,dx\biggr)^{\frac{1}{m}}\biggl( \int_{{\mathbb{R}} ^{N}}\bigl\vert h_{1}(x)\bigr\vert \vert u_{n}\vert ^{m}\,dx\biggr)^{\frac{m-1}{m}} \\ \to&0. \end{aligned}$$
(2.24)
Similarly, we can derive from the limit (ii) in Lemma 2.2 that
$$\begin{aligned} \int_{{\mathbb{R}}^{N}}h_{2}(x)|u_{n}|^{q-1}|u_{n}-v| \, dx \leq&\biggl( \int_{{\mathbb{R}} ^{N}}h_{2}(x)|u_{n}-v|^{q} \,dx\biggr)^{\frac{1}{q}}\biggl( \int_{{\mathbb {R}}^{N}}h_{2}(x)|u_{n}|^{q}\,dx \biggr)^{\frac{q-1}{q}} \\ \to&0. \end{aligned}$$
(2.25)
Then (2.24) and (2.25) show that as \(n\to\infty\)
$$\begin{aligned} o_{n}(1) =&P_{n}-Q_{n} \\ =& \int_{{\mathbb{R}}^{N}}\bigl(\bigl(|\Delta u_{n}|^{p-2} \Delta u_{n}-|\Delta v|^{p-2}\Delta v\bigr) \Delta(u_{n}-v) \\ &{}+\bigl(|\nabla u_{n}|^{p-2}\nabla u_{n}-|\nabla v|^{p-2}\nabla v\bigr) \nabla (u_{n}-v) \\ &{}+V(x) \bigl(|u_{n}|^{p-2}u_{n}-|v|^{p-2}v\bigr) (u_{n}-v)\bigr)\, dx. \end{aligned}$$
(2.26)
Then we have \(\|u_{n}-v\|_{E}\to0\) as \(n\to\infty\). Thus \(J(u)\) satisfies the \((\mathit{PS})\) condition on E and the proof is completed. □

3 Proof of Theorem 1.1

In this section, we will give the proof of Theorem 1.1. We assume that all conditions in the theorem hold. The proof mainly relies on the mountain pass theorem.

Lemma 3.1

([13])

Let E be an infinite dimensional real Banach space, \(J\in C^{1}(E,{\mathbb{R}})\) be even and satisfies the \((\mathit{PS})\) condition and \(J(0)=0\). Assume \(E=Y\oplus Z\), Y is finite dimensional, and J satisfies:
(J1): 

There exist constants \(\rho,\alpha>0\) such that \(J(u)\leq \alpha\) on \(\partial B_{\rho}\cap Z\).

(J2): 

For each finite dimensional subspace \(E_{0}\subset E\), there is an \(R_{0}=R_{0}(E_{0})\) such that \(J(u)\leq0\) on \(E_{0}\setminus B_{R_{0}}\), where \(B_{r}=\{u\in E: \|u\|_{E}< r\}\).

Then J possesses an unbounded sequence of critical values.

Proof of Theorem 1.1

Clearly, the functional J defined by (2.6) is even in E. By Lemma 2.2 in Section 2, the functional satisfies the \((\mathit{PS})\) condition. Next, we prove that J satisfies (J1) and (J2). From (2.9) and (2.10), it follows that
$$ J(u)\geq\frac{1}{p}\|u\|^{p}_{E}-\lambda \frac{a_{1}}{m}\|u\|^{m}_{E}-\frac {a_{2}}{q}\|u \|^{q}_{E}, \quad u\in E. $$
Denote
$$ \phi(z)=z^{p}\biggl(\frac{1}{p}-\lambda\frac{a_{1}}{m}z^{m-p}- \frac {a_{2}}{q}z^{q-p}\biggr),\quad z>0. $$
Then, there exist \(\lambda_{0},z_{1},\alpha>0\) such that \(\phi(z_{1})\geq \alpha\) for any \(\lambda\in[0,\lambda_{0}]\). Let \(\rho=z_{1}\), we have \(J(u)\geq\alpha\) with \(\|u\|_{E}=\rho\) and \(\lambda\in[0,\lambda _{0}]\). So the condition (J1) is satisfied.
We now verify (J2). For any finite dimensional subspace \(E_{0}\subset E\), we assert that there is a constant \(R_{0}>\rho\) such that \(J<0\) on \(E_{0}\setminus B_{R_{0}}\). Otherwise, there exists a sequence \(\{u_{n}\} \subset E_{0}\) such that \(\|u\|_{n}\to\infty\) and \(J(u_{n})\geq0\). Hence
$$ \frac{1}{p}\|u_{n}\|^{p}_{E} \geq\frac{\lambda}{m} \int _{R^{N}}h_{1}|u_{n}|^{m}\,dx+ \frac{1}{q} \int_{{\mathbb{R}}^{N}}h_{2}\|u_{n}\|^{q}\,dx. $$
(3.1)
Set \(\omega_{n}=\frac{u_{n}}{\|u_{n}\|_{E}}\). Then up to a sequence, we can assume \(\omega_{n}\rightharpoonup\omega\) in E, \(\omega_{n}\rightarrow \omega\) a.e. in \({\mathbb{R}}^{N}\). Denote \(\Omega=\{x\in{\mathbb {R}}^{N}:\omega(x)\neq 0\}\). Assume \(|\Omega|>0\). Clearly, \(u_{n}(x)\to\infty\) in Ω. It follows from (2.8) and (2.9) that
$$ \|u\|^{-p}_{E} \int_{\Omega}|h_{1}||u_{n}|^{m}\,dx \leq a_{1}\|u_{n}\|^{m-p}_{E}\to 0 \quad \mbox{as }n\to\infty. $$
On the other hand, we derive
$$ \|u\|^{-p}_{E} \int_{\Omega}|h_{2}||u_{n}|^{q}\,dx \leq a_{2}\|u_{n}\|^{q-p}_{E}\to \infty \quad \mbox{as }n\to\infty. $$
Therefore, multiplying (3.1) by \(\|u\|_{E}^{-p}\) and passing to the limit as \(n\to\infty\) show that \(\frac{1}{p}\geq\infty\). This is impossible. So \(|\Omega|=0\) and \(\omega(x)=0\) a.e. on \({\mathbb {R}}^{N}\). By the equivalence of all norms in \(E_{0}\), there exists a constant \(\beta >0\) such that
$$ \int_{{\mathbb{R}}^{N}}|h_{2}||u|^{q}\,dx\geq \beta^{q}\|u\|^{q}_{E}, \quad \forall u\in E_{0} \quad \mbox{and}\quad \int_{{\mathbb{R}}^{N}}|h_{2}||u_{n}|^{q}\,dx \geq\beta^{q}\|u_{n}\| ^{q}_{E}, \quad \forall n\in{\mathbb{N}}. $$
Hence
$$ 0< \beta^{q}\leq\lim\sup_{n\to\infty} \int_{{\mathbb{R}} ^{N}}|h_{2}||\omega_{n}|^{q} \,dx\leq \int_{{\mathbb{R}}^{N}}\limsup_{n\to \infty}\bigl(|h_{2}|| \omega_{n}|^{q}\bigr)\, dx= \int_{{\mathbb{R}}^{N}}\bigl(|h_{2}||\omega|^{q}\bigr)\, dx=0. $$
This is a contradiction. So there exists a constant \(R_{0}\) such that \(J<0\) on \(E_{0}\setminus B_{R_{0}}\). Therefore, the existence of infinitely many solutions \(\{u_{n}\}\) for problem (1.1) follows from Lemma 3.1 and we finish the proof of Theorem 1.1. □

Declarations

Acknowledgements

The authors wish to express their gratitude to the referees for valuable comments and suggestions. This work was supported by the Project of Innovation in Scientific Research for Graduate Students of Jiangsu Province (No. CXZZ13-0263) and the Fundamental Research Funds for the Central Universities of China (2015B31014).

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)
College of Science, Hohai University

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