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  • Research
  • Open Access

High energy solutions of modified quasilinear fourth-order elliptic equation

Boundary Value Problems20182018:54

https://doi.org/10.1186/s13661-018-0970-6

©  The Author(s) 2018

  • Received: 21 September 2017
  • Accepted: 28 March 2018
  • Published:

Abstract

This paper focuses on the following modified quasilinear fourth-order elliptic equation:
$$\textstyle\begin{cases} \triangle^{2}u-(a+b\int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\triangle u+\lambda V(x)u-\frac{1}{2}\triangle(u^{2})u=f(x,u),& \mbox{in }\mathbb{R}^{3}, \\ u(x)\in H^{2}(\mathbb{R}^{3}), \end{cases} $$
where \(\triangle^{2}=\triangle(\triangle)\) is the biharmonic operator, \(a>0\), \(b\geq 0\), \(\lambda\geq 1\) is a parameter, \(V\in C(\mathbb{R}^{3},\mathbb{R})\), \(f(x,u)\in C(\mathbb{R}^{3}\times\mathbb{R}, \mathbb{R})\). \(V(x)\) and \(f(x,u)u\) are both allowed to be sign-changing. Under the weaker assumption \(\lim_{ \vert t \vert \rightarrow\infty}\frac{\int^{t}_{0}f(x,s)\,ds}{ \vert t \vert ^{3}}=\infty\) uniformly in \(x\in\mathbb{R}^{3}\), a sequence of high energy weak solutions for the above problem are obtained.

Keywords

  • Super-quadratic
  • High energy solutions
  • Sign-changing potential
  • Fountain theorem

MSC

  • 35J25
  • 35J20
  • 35J60
  • 35J61

1 Introduction and main results

In this paper, we consider the following elliptic equation:
$$ \textstyle\begin{cases} \triangle^{2}u-(a+b\int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\triangle u+\lambda V(x)u-\frac{1}{2}\triangle(u^{2})u=f(x,u),& \mbox{in } \mathbb{R}^{3}, \\ u(x)\in H^{2}(\mathbb{R}^{3}) , \end{cases} $$
(1.1)
where \(\triangle^{2}=\triangle(\triangle)\) is the biharmonic operator, the constants \(a>0\), \(b\geq 0\), and \(\lambda\geq 1\) is a parameter. \(V(x):\mathbb{R}^{3}\rightarrow \mathbb{R}\) and \(f:\mathbb{R}^{3}\times\mathbb{R}\rightarrow\mathbb{R}\) satisfies the following assumptions:
\((V)\) : 
\(V\in C(\mathbb{R}^{3},\mathbb{R})\), \(\inf_{\mathbb{R}^{3}}V>-\infty\) and there exists a constant \(r>0\) such that
$$\lim_{ \vert y \vert \rightarrow\infty}\operatorname{meas}\bigl\{ x\in\mathbb{R}^{3}: \vert x-y \vert \leq r, V(x)\leq M\bigr\} =0,\quad \forall M>0; $$
\((F_{1})\) : 
\(f\in C(\mathbb{R}^{3}\times\mathbb{R},\mathbb{R})\) and there exists positive constant \(C_{0}\) and \(p>4\) such that
$$\bigl\vert f(x,t) \bigr\vert \leq C_{0}\bigl( \vert t \vert + \vert t \vert ^{p-1}\bigr),\quad \forall(x,t)\in \mathbb{R}^{3}\times\mathbb{R}. $$
\((F_{2})\) : 

\(\lim_{ \vert t \vert \rightarrow\infty}\frac{F(x,t)}{ \vert t \vert ^{3}}=\infty\) uniformly in \(x\in\mathbb{R}^{3}\), where \(F(x,t)=\int^{t}_{0}f(x,s)\,ds\).

\((F_{3})\) : 
There exists a constant \(\alpha\geq 0\) such that
$$f(x,t)t-4F(x,t)\geq-\alpha t^{2}, \quad \forall(x,t)\in \mathbb{R}^{3}\times\mathbb{R}. $$
\((F_{4})\) : 

\(f(x,-t)=-f(x,t)\) for all \((x,t)\in\mathbb{R}^{3}\times\mathbb{R}\).

The Kirchhoff’s model considers the changes in length of the string produced by transverse vibrations. It was pointed out in [1–4] that (1.1) models several physical and biological systems where u describes a process which relies on the mean of itself such as the population density. For more mathematical and physical background on Kirchhoff-type problems, we refer the reader to [1, 5–8] and the references therein. It is well known that fourth-order elliptic equation has been widely studied since Lazer and Mckenna [9] first proposed to study periodic oscillations and traveling waves in a suspension bridge.

In te recent years, many scholars widely studied the Schrödinger equation under variant assumptions on \(V(x)\) and \(f(x,u)\), such as [3, 4, 10–13]. In [10], Wu considered the following Schrödinger–Kirchhoff-type problem:
$$ -\biggl(a+b \int_{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx\biggr) \triangle u+V(x)u=f(x,u),\quad \mbox{in } \mathbb{R}^{N}(N\leq3) $$
(1.2)
under these hypotheses:
\((V')\) : 

\(V\in C(\mathbb{R}^{N},\mathbb{R})\) satisfies \(\inf V(x)\geq a_{1}>0\) and for each \(M>0\), \(\operatorname{meas}\{x\in\mathbb{R}^{N}:V(x)\leq M\}<+\infty\), where \(a_{1}\) is a constant and meas denotes the Lebesgue measure in \(\mathbb{R}^{N}\).

\((f_{1})\) : 

\(f\in C(\mathbb{R}^{N}\times\mathbb{R},\mathbb{R})\) and \(\vert f(x,t) \vert \leq C(1+ \vert t \vert ^{p-1})\) for some \(2\leq p<2^{\ast}\), where C is a positive constant;

\((f_{2})\) : 

\(f(x,t)=o( \vert t \vert )\) as \(\vert t \vert \rightarrow 0\);

\((f_{3})\) : 

\(\frac{F(x,t)}{t^{4}}\rightarrow+\infty\) as \(\vert t \vert \rightarrow+\infty\) uniformly in \(\forall x\in\mathbb{R}^{N}\);

\((f_{4})\) : 

\(tf(x,t)\geq4F(x,t)\), \(\forall x\in\mathbb{R}^{N}\), \(\forall t\in\mathbb{R}\).

Here \((f_{3})\) is essential in these references to overcome the missing of compactness. The author got a nontrivial solution of (1.2). In [8], Zhang and Tang also considered the problem (1.2) under the assumption \((V)\), and they obtained infinitely many high energy solutions of the problem (1.2). In [11], Nie studied the following Schrödinger–Kirchhoff-type equation:
$$ \textstyle\begin{cases} -(a+b\int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\triangle u+\lambda V(x)u=f(x,u),& \mbox{in }\mathbb{R}^{3}, \\ u(x)\rightarrow 0& \mbox{as } \vert x \vert \rightarrow\infty , \end{cases} $$
(1.3)
under the assumption \((V')\). They got a sequence of high energy weak solutions whenever \(\lambda>0\) is sufficiently large. In [14], Xu and Chen also used condition \((V')\) to study the problem (1.3).
More recently, Cheng and Tang [15] studied the following elliptic equation:
$$ \textstyle\begin{cases} \triangle^{2}u-\triangle u+V(x)u-\frac{1}{2}\triangle(u^{2})u=f(x,u),& \mbox{in }\mathbb{R}^{N} \\ u(x)\in H^{2}(\mathbb{R}^{N}) , \end{cases} $$
(1.4)
under the assumption \((f_{3})\). Clearly, the problem (1.1) is equivalent to (1.4) whenever \(N=3\), \(a=1\), \(b=0\), \(\lambda=1\), and condition \((f_{3})\) is stronger than \((F_{2})\).

Motivated by the work we discussed above, we will use weaker conditions \((F_{2})\), \((F_{3})\) instead of the common assumptions \((f_{3})\), \((f_{4})\), while \(V(x)\) and \(f(x,u)u\) are both allowed to be sign-changing. We will further study and establish the existence of infinitely many high energy solutions of (1.1) whenever \(\lambda\geq 1\), by using the fountain theorem [16, 17] or its other versions [18, 19]. To the best of our knowledge, there is little work concerning this case up to now.

The following are our main results.

Theorem 1.1

Assume that \((V)\) and \((F_{1})\)–\((F_{4})\) are satisfied, then problem (1.1) possesses infinitely many high energy solutions whenever \(\lambda\geq 1\).

Corollary 1.2

Assume that \((V)\) and \((F_{1})\)–\((F_{4})\) are satisfied, then problem (1.3) possesses infinitely many high energy nontrivial solutions whenever \(\lambda\geq 1\).

Remark 1.3

Obviously, the condition \((V)\) is weaker than \((V')\); \((F_{1})\) is weaker than \((f_{1})\) and \((f_{2})\); \((F_{3})\) is weaker than \((f_{7})\) [14] and \((f_{4})\); \((F_{2})\) is weaker than \((g_{2})\) [15]. Furthermore, we do not require λ large enough, but we only need \(\lambda\geq 1\). Therefore, our results extend and improve Theorem 1 [10], Theorem 1.2 [11], Theorem 1.3 [14], Theorem 1.1 [8], Theorem 1.4 [15] and so on.

Remark 1.4

There are many functions satisfying assumptions \((F_{1})\)–\((F_{4})\) not \((f_{3})\). For example
$$f(x,u)=4u^{3}-\frac{2u(1+u^{2})\ln(1+u^{2})+2u^{3}-2u^{3}\ln(1+u^{2})}{(1+u^{2})^{2}} $$
for all \((x,u)\in\mathbb{R}^{3}\times\mathbb{R}\).
Indeed, \(F(x,u)=u^{4}-\frac{u^{2}\ln(1+u^{2})}{1+u^{2}}\), then we can find a positive constant α such that
$$f(x,u)u-4F(x,u)+\alpha u^{2}=\frac{2u^{2}\ln(1+u^{2}-2u^{4}+\alpha u^{6}+2\alpha u^{4}+\alpha u^{2})}{(1+u^{2})^{2}}\geq0. $$

2 Preliminary lemmas and proof of our main result

In order to apply the variational method, we first recall some related preliminaries and establish a corresponding variational framework for our problem (1.1); then we give the proof of Theorem 1.1.

For \(1< s<+\infty\), define the Sobolev space
$$W^{m,s}\bigl(\mathbb{R}^{N}\bigr)=\bigl\{ u\in L^{s}\bigl(\mathbb{R}^{N}\bigr)\mid D^{\alpha}u\in L^{s}\bigl(\mathbb{R}^{N}\bigr), \vert \alpha \vert \leq m\bigr\} $$
equipped with the norm
$$\Vert u \Vert _{W^{m,s}(\mathbb{R}^{N})}= \biggl(\sum_{ \vert \alpha \vert \leq m} \int_{\mathbb{R}^{N}} \bigl\vert D^{\alpha}u \bigr\vert ^{s}\,dx \biggr)^{\frac{1}{s}}, $$
where \(\alpha=(\alpha_{1}, \alpha_{2},\ldots, \alpha_{N})\) with \(\alpha_{i}\in \mathbb{Z}^{+} \) (the set of all non-negative integers), \(i=1, 2, \ldots, N\), \(\vert \alpha \vert =\alpha_{1}+\alpha_{2}+\cdots+\alpha_{N}\) and
$$D^{\alpha}u=\frac{\partial^{ \vert \alpha \vert }u}{\partial x^{\alpha_{1}}_{1}\partial x^{\alpha_{2}}_{2}\cdots\partial x^{\alpha_{N}}_{N}}. $$
For \(s=2\), \(H^{m}(\mathbb{R}^{N})=W^{m,2}(\mathbb{R}^{N})\) is a Hilbert space equipped with the scalar product
$$\langle u,v\rangle_{H^{m}}=\sum_{ \vert \alpha \vert \leq m} \int_{\mathbb{R}^{N}}D^{\alpha}u D^{\alpha}v\,dx $$
and the norm
$$\Vert u \Vert _{H^{m}}=\langle u,u\rangle_{H^{m}}^{\frac{1}{2}}= \biggl(\sum_{ \vert \alpha \vert \leq m} \int_{\mathbb{R}^{N}} \bigl\vert D^{\alpha}u \bigr\vert ^{2}\,dx \biggr)^{\frac{1}{2}}. $$
Moreover, for \(m=2\) one has
$$\begin{aligned} &\langle u,v\rangle_{H^{2}}= \int_{\mathbb{R}^{N}}(\triangle u\triangle v+\nabla u \nabla v+uv)\,dx, \\ & \Vert u \Vert ^{2}_{H^{2}}=\langle u,v \rangle_{H^{2}}= \int_{\mathbb{R}^{N}}\bigl( \vert \triangle u \vert ^{2}+ \vert \nabla u \vert ^{2}+u^{2}\bigr)\,dx, \end{aligned}$$
whenever \(u,v \in H^{2}(\mathbb{R}^{N})\).
Under assumption \((V)\), we can find \(V_{0}\geq 0\) such that \(\widetilde{V}(x)=V(x)+V_{0}\geq 1\) for all \(x\in\mathbb{R}^{3}\). Then
$$E_{\lambda}= \biggl\{ u\in H^{2}\bigl(\mathbb{R}^{3} \bigr): \int_{\mathbb{R}^{3}}\bigl(a \vert \nabla u \vert ^{2}+ \lambda\widetilde{V}(x)u^{2}\bigr)\,dx< \infty \biggr\} $$
is a Hilbert space endowed with the norm
$$\Vert u \Vert _{\lambda}= \biggl( \int_{\mathbb{R}^{3}}\bigl( \vert \triangle u \vert ^{2}+a \vert \nabla u \vert ^{2}+\lambda\widetilde{V}(x)u^{2} \bigr)\,dx \biggr)^{\frac{1}{2}}. $$
Let
$$\begin{aligned} \Phi_{\lambda}(u) =&\frac{1}{2} \int_{\mathbb{R}^{3}}\bigl( \vert \triangle u \vert ^{2}+a \vert \nabla u \vert ^{2}+\lambda V(x)u^{2}\bigr)\,dx+ \frac{b}{4} \biggl( \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \biggr)^{2} \\ &{}+\frac{1}{2} \int_{\mathbb{R}^{3}}u^{2} \vert \nabla u \vert ^{2}\,dx- \int_{\mathbb{R}^{3}}F(x,u)\,dx,\quad \forall u\in E_{\lambda}. \end{aligned}$$
(2.1)
By condition \((V)\), \((F_{1})\) and the fact \(\int_{\mathbb{R}^{3}}u^{2} \vert \nabla u \vert ^{2}\,dx<\infty\) (see Lemma 2.2 in [20]), \(\Phi_{\lambda}\) is a well-defined class \(C^{1}\) functional. For all \(u,v\in E_{\lambda}\)
$$\begin{aligned} \bigl\langle \Phi'_{\lambda}(u),v\bigr\rangle =& \int_{\mathbb{R}^{3}}\bigl(\triangle u\triangle v+a\nabla u\nabla v+\lambda V(x)uv\bigr)\,dx+b \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \int_{\mathbb{R}^{3}}\nabla u\nabla v\,dx \\ &{}+ \int_{\mathbb{R}^{3}}\bigl(uv \vert \nabla u \vert ^{2}+u^{2} \nabla u \nabla v\bigr)\,dx- \int_{\mathbb{R}^{3}}f(x,u)v\,dx. \end{aligned}$$
(2.2)
Clearly, seeking a weak solution of problem (1.1) is equivalent to finding a critical point of the functional \(\Phi_{\lambda}\).

Definition 2.1

A sequence \(\{u_{n}\}\subset E_{\lambda}\) is said to be a \((C)_{c}\) sequence if
$$\Phi_{\lambda}(u_{n})\rightarrow c,\qquad \bigl\Vert \Phi'_{\lambda}(u_{n}) \bigr\Vert _{\lambda} \bigl(1+ \Vert u_{n} \Vert _{\lambda}\bigr)\rightarrow 0. $$
\(\Phi_{\lambda}\) is said to satisfy the \((C)_{c}\) condition if any \((C)_{c}\) sequence possesses a convergent subsequence.

Let \(E'_{\lambda}= \{u\in H^{2}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}(a \vert \nabla u \vert ^{2}+\lambda\widetilde{V}(x)u^{2})\,dx<\infty \}\).

Lemma 2.2

Under assumption \((V)\), the embedding \(E'_{\lambda}\hookrightarrow L^{s}(\mathbb{R}^{N})\) is compact for \(2\leq s<2_{\ast}\), where \(2_{\ast}=\frac{2N}{N-4}\), if \(N>4\); \(2_{\ast}=+\infty\), if \(N\leq4\).

Proof

Define
$$E= \biggl\{ u\in H^{1}\bigl(\mathbb{R}^{N}\bigr): \int_{\mathbb{R}^{N}}\bigl(a \vert \nabla u \vert ^{2}+ \lambda\widetilde{V}(x)u^{2}\bigr)\,dx< \infty \biggr\} . $$
By Propositions 3.1 and 3.3 in [13], we know that the embedding \(E\hookrightarrow L^{s}(\mathbb{R}^{N})\) is compact for \(2\leq s<2_{\ast}\) due to the condition \((V)\), and the embedding \(E'_{\lambda}\hookrightarrow E\) is continuous, therefore, the embedding \(E'_{\lambda}\hookrightarrow L^{s}(\mathbb{R}^{N})\) is compact for \(2\leq s<2_{\ast}\). □

Lemma 2.3

Under assumptions \((V)\), \((F_{1})\), any bounded \((C)_{c}\) sequence of \(\Phi_{\lambda}\) has a strongly convergent subsequence in \(E_{\lambda}\).

Proof

Let \(\{u_{n}\}\subset E_{\lambda}\) hold with
$$ \sup_{n} \Vert u_{n} \Vert _{\lambda}< +\infty. $$
(2.3)
Then up to a subsequence, there exists a constant \(c\in\mathbb{R}\) such that
$$ \Phi_{\lambda}(u_{n})\rightarrow c,\qquad \Phi'_{\lambda}(u_{n})\rightarrow 0. $$
(2.4)
According to Lemma 2.2, going if necessary to a subsequence, we can assume that there exists \(u\in E_{\lambda}\) such that
$$\begin{aligned} &u_{n}\rightharpoonup u\quad \mbox{in }E_{\lambda}, \\ &u_{n}\rightarrow u \quad \mbox{in }L^{s}\bigl( \mathbb{R}^{3}\bigr)\ (2\leq s< +\infty), \\ &u_{n}\rightarrow u\quad \mbox{a.e. in } \mathbb{R}^{3}. \end{aligned}$$
(2.5)
By an elementary computation,
$$\begin{aligned} &\bigl\langle \Phi'_{\lambda}(u_{n})- \Phi'(u),u_{n}-u\bigr\rangle \\ &\quad \geq \Vert u_{n}-u \Vert ^{2}_{\lambda}- \lambda V_{0} \int_{\mathbb{R}^{3}} \vert u_{n}-u \vert ^{2}\,dx \\ &\qquad {}+b\biggl( \int_{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{2}\,dx- \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx\biggr) \int_{\mathbb{R}^{3}}\nabla u_{n} \nabla (u_{n}-u) \,dx \\ &\qquad {}+ \int_{\mathbb{R}^{3}}\bigl(u_{n} \vert \nabla u_{n} \vert ^{2}-u \vert \nabla u \vert ^{2}\bigr) (u_{n}-u)\,dx + \int_{\mathbb{R}^{3}}\bigl(u^{2}_{n}-u^{2} \bigr)\nabla u\nabla (u_{n}-u)\,dx \\ &\qquad {}+ \int_{\mathbb{R}^{3}}\bigl(f(x,u)-f(x,u_{n})\bigr) (u_{n}-u)\,dx. \end{aligned}$$
(2.6)
Clearly, \(\lambda V_{0}\int_{\mathbb{R}^{3}} \vert u_{n}-u \vert ^{2}\,dx\rightarrow 0\), and \(\langle\Phi'_{\lambda}(u_{n})-\Phi'(u),u_{n}-u\rangle\rightarrow 0\). Then, since \(\{u_{n}\}\subset E_{\lambda}\) is bounded, we have
$$\begin{aligned} & \biggl\vert b\biggl( \int_{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{2}\,dx- \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx\biggr) \int_{\mathbb{R}^{3}}\nabla u_{n}\nabla (u_{n}-u)\,dx \biggr\vert \\ &\quad \leq \biggl\vert b\biggl( \int_{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{2}\,dx- \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx\biggr) \int_{\mathbb{R}^{3}}\nabla u\nabla (u_{n}-u)\,dx \biggr\vert \\ &\qquad {}+ \biggl\vert b\biggl( \int_{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{2}\,dx- \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx\biggr) \int_{\mathbb{R}^{3}} \bigl\vert \nabla (u_{n}-u) \bigr\vert ^{2}\,dx \biggr\vert \\ &\quad \rightarrow 0. \end{aligned}$$
(2.7)
Note that \(E_{\lambda}\hookrightarrow H^{2}(\mathbb{R}^{3})\hookrightarrow W^{1,s}(\mathbb{R}^{3})\) for \(2\leq s \leq+\infty\),
$$\begin{aligned} \int_{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{3}\,dx &\leq \int_{\mathbb{R}^{3}} \Biggl( \vert u_{n} \vert ^{2}+\sum^{3}_{i=1} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{2} \Biggr)^{\frac{3}{2}}\,dx \\ &\leq \int_{\mathbb{R}^{3}} \Biggl( \vert u_{n} \vert +\sum ^{3}_{i=1} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert \Biggr)^{3}\,dx \\ &\leq \int_{\mathbb{R}^{3}} \biggl[4\max \biggl\{ \vert u_{n} \vert , \biggl\vert \frac{\partial u_{n}}{\partial x_{1}} \biggr\vert , \biggl\vert \frac{\partial u_{n}}{\partial x_{2}} \biggr\vert , \biggl\vert \frac{\partial u_{n}}{\partial x_{3}} \biggr\vert \biggr\} \biggr]^{3}\,dx \\ &\leq4^{3} \int_{\mathbb{R}^{3}} \Biggl( \vert u_{n} \vert ^{3}+\sum^{3}_{i=1} \biggl\vert \frac{\partial u_{n}}{\partial x_{i}} \biggr\vert ^{3} \Biggr)\,dx \\ &=4^{3} \Vert u_{n} \Vert ^{3}_{W^{1,3}(\mathbb{R}^{3})} \\ &\leq4^{3}S^{3}_{3} \Vert u_{n} \Vert ^{3}_{\lambda}, \end{aligned}$$
(2.8)
where
$$S_{s}=\sup_{u\in E_{\lambda}, \Vert u \Vert _{\lambda}=1} \Vert u \Vert _{W^{1,s}}, \quad \forall 2\leq s\leq +\infty. $$
Applying (2.3)–(2.5) and (2.8), there exist constants \(C_{1}>0\) such that
$$\begin{aligned} & \biggl\vert \int_{\mathbb{R}^{3}}\bigl(u_{n} \vert \nabla u_{n} \vert ^{2}-u \vert \nabla u \vert ^{2}\bigr) (u_{n}-u)\,dx \biggr\vert \\ &\quad \leq \int_{\mathbb{R}^{3}} \vert u_{n} \vert \vert \nabla u_{n} \vert ^{2} \vert u_{n}-u \vert \,dx+ \int_{\mathbb{R}^{3}} \vert u \vert \vert \nabla u \vert ^{2} \vert u_{n}-u \vert \,dx \\ &\quad \leq \biggl( \int_{\mathbb{R}^{3}} \vert u_{n} \vert ^{6}\,dx \biggr)^{\frac{1}{6}} \biggl( \int_{\mathbb{R}^{3}} \vert \nabla u_{n} \vert ^{3}\,dx \biggr)^{\frac{2}{3}} \biggl( \int_{\mathbb{R}^{3}} \vert u_{n}-u \vert ^{6} \,dx \biggr)^{\frac{1}{6}} \\ &\qquad {}+ \biggl( \int_{\mathbb{R}^{3}} \vert u \vert ^{6}\,dx \biggr)^{\frac{1}{6}} \biggl( \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{3}\,dx \biggr)^{\frac{2}{3}} \biggl( \int_{\mathbb{R}^{3}} \vert u_{n}-u \vert ^{6} \,dx \biggr)^{\frac{1}{6}} \\ &\quad \leq C_{1} \Vert u_{n}-u \Vert _{L^{6}} \rightarrow 0, \quad \mbox{as } n\rightarrow\infty, \end{aligned}$$
(2.9)
and \(C'_{1}>0\) such that
$$\begin{aligned} & \biggl\vert \int_{\mathbb{R}^{3}}\bigl(u_{n}^{2}-u^{2} \bigr)\nabla u\nabla(u_{n}-u)\,dx \biggr\vert \\ &\quad \leq \int_{\mathbb{R}^{3}} \vert u_{n}-u \vert \vert u_{n}+u \vert \vert \nabla u \vert \bigl\vert \nabla(u_{n}-u) \bigr\vert \,dx \\ &\quad \leq \biggl( \int_{\mathbb{R}^{3}} \vert u_{n}-u \vert ^{6} \biggr)^{\frac{1}{6}} \biggl( \int_{\mathbb{R}^{3}} \vert u_{n}+u \vert ^{6} \biggr)^{\frac{1}{6}} \biggl( \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{3} \biggr)^{\frac{1}{3}} \biggl( \int_{\mathbb{R}^{3}} \bigl\vert \nabla(u_{n}-u) \bigr\vert ^{3} \biggr)^{\frac{1}{3}} \\ &\quad \leq C'_{1} \Vert u_{n}-u \Vert _{L^{6}}\rightarrow 0, \quad \mbox{as }n\rightarrow\infty. \end{aligned}$$
(2.10)
By \((F_{1})\) and the Hölder inequality,
$$\begin{aligned} & \biggl\vert \int_{\mathbb{R}^{3}}\bigl(f(x,u)-f(x,u_{n})\bigr) (u_{n}-u)\,dx \biggr\vert \\ &\qquad \leq C_{0} \int_{\mathbb{R}^{3}}\bigl[ \vert u \vert + \vert u \vert ^{p-1}+ \vert u_{n} \vert + \vert u_{n} \vert ^{p-1}\bigr] \vert u_{n}-u \vert \,dx \\ &\qquad \leq C_{0}\bigl[\bigl( \Vert u_{n} \Vert _{L^{2}}+ \Vert u \Vert _{L^{2}}\bigr) \Vert u_{n}-u \Vert _{L^{2}}+\bigl( \Vert u_{n} \Vert ^{p-1}_{L^{p}}+ \Vert u \Vert ^{p-1}_{L^{p}} \bigr) \Vert u_{n}-u \Vert _{L^{p}}\bigr]. \end{aligned}$$
Then, combining the last inequality with (2.5), we get
$$ \int_{\mathbb{R}^{3}}\bigl(f(x,u)-f(x,u_{n})\bigr) (u_{n}-u)\,dx\rightarrow 0,\quad \mbox{as }n\rightarrow\infty. $$
(2.11)
Hence, the combination of (2.7) and (2.9)–(2.11) implies that
$$u_{n}\rightarrow u \quad \mbox{in }E_{\lambda}. $$
Therefore, the proof is complete. □

Lemma 2.4

Assume that \((V)\) and \((F_{1})\)–\((F_{3})\) hold, then \(\Phi_{\lambda}\) satisfies the \((C)_{c}\) condition.

Proof

Let \(\{u_{n}\}\subset E_{\lambda}\) be such that
$$ \Phi_{\lambda}(u_{n})\rightarrow c,\qquad \bigl\Vert \Phi'_{\lambda}(u_{n}) \bigr\Vert _{\lambda}\bigl(1+ \Vert u_{n} \Vert _{\lambda}\bigr) \rightarrow 0. $$
(2.12)
First, we prove that \(\{u_{n}\}\) is bounded in \(E_{\lambda}\). By \((F_{3})\), (2.1), (2.2) and (2.12), one has
$$\begin{aligned} c+o(1) =&\Phi_{\lambda}(u_{n})-\frac{1}{4} \bigl\langle \Phi'_{\lambda}(u_{n}),u_{n} \bigr\rangle \\ =&\frac{1}{4} \int_{\mathbb{R}^{3}}\bigl( \vert \triangle u_{n} \vert ^{2}+a \vert \nabla u_{n} \vert ^{2}+\lambda \widetilde{V}(x)u^{2}_{n}\bigr)\,dx \\ &{}+ \int_{\mathbb{R}^{3}} \biggl[\frac{1}{4}f(x,u_{n})u_{n}-F(x,u_{n})- \frac{\lambda}{4}V_{0}u^{2}_{n} \biggr]\,dx \\ \geq&\frac{1}{4} \Vert u_{n} \Vert ^{2}_{\lambda}- \frac{\alpha+\lambda V_{0}}{4} \int_{\mathbb{R}^{3}}u^{2}_{n}\,dx. \end{aligned}$$
(2.13)
Thus, it remains to show that \(\{u_{n}\}\) is bounded in \(L^{2}(\mathbb{R}^{3})\). Otherwise, suppose that \(\Vert u_{n} \Vert _{2}\rightarrow\infty\) and then \(\Vert u_{n} \Vert _{\lambda}\rightarrow\infty\). Let \(\omega_{n}=\frac{u_{n}}{ \Vert u_{n} \Vert _{\lambda}}\), then \(\Vert \omega_{n} \Vert _{\lambda}=1\). According to Lemma 2.2, up to a subsequence, for some \(\omega\in E_{\lambda}\), we obtain
$$\begin{aligned} &\omega_{n} \rightharpoonup\omega \quad \mbox{in } E_{\lambda}, \\ &\omega_{n} \rightarrow\omega \quad \mbox{in } L^{2}\bigl( \mathbb{R}^{3}\bigr), \\ &\omega_{n} \rightarrow\omega \quad \mbox{a.e. in } \mathbb{R}^{3}. \end{aligned}$$
Clearly, we deduce that \(\omega\neq 0\) from (2.13). Then, for \(x\in\{y\in\mathbb{R}^{3}:\omega(y)\neq 0\}\), we have \(\vert u_{n}(x) \vert \rightarrow\infty\) as \(n\rightarrow\infty\). For any given \(u\in H^{2}(\mathbb{R}^{3})\backslash\{0\}\), define
$$\begin{aligned} g(t)&= \bigl\Vert t^{-1}u(tx) \bigr\Vert ^{2}_{H^{2}}-1 \\ &=\frac{1}{t} \int_{\mathbb{R}^{3}} \vert \triangle u \vert ^{2}\,dx+ \frac{1}{t^{3}} \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx+ \frac{1}{t^{5}} \int_{\mathbb{R}^{3}}u^{2}\,dx-1, \quad \forall t>0. \end{aligned}$$
By an elementary computation, there exists a unique \(T=\widetilde{t}(u)>0\) such that
$$g(T)=0, \quad \forall u\in H^{2}\bigl(\mathbb{R}^{3}\bigr) \backslash\{0\}. $$
This implies that \(g(t)=0\) defines a functional \(T=\widetilde{t}(u)\) on \(H^{2}(\mathbb{R}^{3})\backslash\{0\}\). We define \(\widetilde{t}(0)=0\). It is easy to verify that \(T=\widetilde{t}(u)\) is continuous and \(\widetilde{t}(u)\rightarrow\infty\) as \(\Vert u \Vert _{H^{2}}\rightarrow\infty\).
Due to the definition of g, for any \(u\in H^{2}(\mathbb{R}^{3})\backslash\{0\}\), there exists
$$v(x)=T^{-1}u(Tx)\in H^{2}\bigl(\mathbb{R}^{3} \bigr) $$
such that
$$\Vert v \Vert _{H^{2}}=1. $$
Note that \(u_{n}\neq 0\) for large \(n\in\mathbb{N}\), then there exist
$$v_{n}(x)=T_{n}^{-1}u_{n}(T_{n}x) \in H^{2}\bigl(\mathbb{R}^{3}\bigr) $$
such that
$$\Vert v_{n} \Vert _{H^{2}}=1. $$
That is,
$$u_{n}(x)=T_{n} v_{n}\bigl(T^{-1}_{n}x \bigr), $$
with \(\Vert v_{n} \Vert _{H^{2}}=1\) for large \(n\in\mathbb{N}\). Moreover, we have
$$T_{n}=\widetilde{t}(u_{n})\rightarrow\infty\quad \mbox{as }n\rightarrow\infty $$
and
$$\bigl\{ x\in\mathbb{R}^{3}:v_{n}(x)\neq0\bigr\} \neq\emptyset \quad \mbox{for large }n\in\mathbb{N}. $$
From \((F_{1})\)–\((F_{3})\), there are \(R_{0}>0\) and \(C_{2}>0\) such that, for all \(x\in\mathbb{R}^{3}\),
$$ f(x,u)u+\alpha u^{2}\geq 4F(x,u)\geq 0, \quad \forall \vert u \vert \geq R_{0}, $$
(2.14)
and
$$ \bigl\vert f(x,u)u \bigr\vert \leq C_{2}u^{2}, \quad \forall \vert u \vert \leq R_{0}. $$
(2.15)
Thus, by \((F_{3})\), (2.1), (2.2), (2.12)–(2.15) and \(\Vert v_{n} \Vert _{H^{2}}=1\),
$$\begin{aligned} c+o(1) =&\Phi_{\lambda}(u_{n})-\frac{1}{2} \bigl\langle \Phi'_{\lambda}(u_{n}),u_{n} \bigr\rangle \\ \geq&-\frac{b}{4} \Vert \nabla u_{n} \Vert ^{4}_{2}-\frac{\alpha}{4} \int_{\mathbb{R}^{3}}u^{2}_{n}\,dx-\frac{1}{2} \int_{\mathbb{R}^{3}}u_{n}^{2} \vert \nabla u_{n} \vert ^{2}\,dx+\frac{1}{4} \int_{\mathbb{R}^{3}}f(x,u_{n})u_{n}\,dx \\ =&-\frac{bT^{6}_{n}}{4} \Vert \nabla v_{n} \Vert ^{4}_{2}-\frac{\alpha T^{5}_{n}}{4} \int_{\mathbb{R}^{3}}v^{2}_{n}\,dx-\frac{T^{5}_{n}}{2} \int_{\mathbb{R}^{3}}v_{n}^{2} \vert \nabla v_{n} \vert ^{2}\,dx \\ &{}+\frac{T^{3}_{n}}{4} \int_{ \vert T_{n}v_{n} \vert \leq R_{0}}f(T_{n}x,T_{n}v_{n})T_{n}v_{n} \,dx +\frac{T^{6}_{n}}{4} \int_{ \vert T_{n}v_{n} \vert \geq R_{0}}\frac{f(T_{n}x,T_{n}v_{n})T_{n}v_{n}}{T^{3}_{n}}\,dx \\ \geq&\frac{T^{6}_{n}}{4} \biggl\{ -b-\frac{\alpha+C_{2}}{T_{n}}+ \int_{ \vert T_{n}v_{n} \vert \geq R_{0}}\frac{f(T_{n}x,T_{n}v_{n})T_{n}v_{n}}{T^{3}_{n}}\,dx \\ \ &{} -\frac{2\int_{\mathbb{R}^{3}}v_{n}^{2} \vert \nabla v_{n} \vert ^{2}\,dx}{T_{n}} \biggr\} . \end{aligned}$$
(2.16)
By the Hölder inequality and the Sobolev embedding inequality, we see that the sequence of integrals \(\int_{\mathbb{R}^{3}}v_{n}^{2} \vert \nabla v_{n} \vert ^{2}\,dx<\infty\), since \(\Vert v_{n} \Vert _{H^{2}}=1\); on the other hand, by \((F_{2})\) and (2.14), we have
$$\int_{ \vert T_{n}v_{n} \vert \geq R_{0}}\frac{f(T_{n}x,T_{n}v_{n})T_{n}v_{n}}{T^{3}_{n}}\,dx\rightarrow+\infty\quad \mbox{as }n\rightarrow+\infty, $$
which contradicts (2.16). Hence, \(\{u_{n}\}\) is bounded in \(L^{2}(\mathbb{R}^{3})\). This shows that \(\{u_{n}\}\) is bounded in \(E_{\lambda}\) due to (2.13). By Lemma 2.3, \(\{u_{n}\}\) contains a convergent subsequence. □
Next, we define
$$X_{j}=\mathbb{R}e_{j}, \qquad Y_{k}= \bigoplus_{j=1}^{k}X_{j}, \qquad Z_{k}= \overline{\bigoplus_{j=k+1}^{\infty}X_{j}}, \quad k\in \mathbb{Z}, $$
where \(\{e_{j}\}\) is an orthonormal basis of \(E_{\lambda}\).

Lemma 2.5

Assume that \((V)\) holds, then, for \(2\leq s<2_{\ast}\),
$$\beta_{k}(s)=\sup_{u\in Z_{k}, \Vert u \Vert _{\lambda}=1} \Vert u \Vert _{s}\rightarrow 0,\quad k\rightarrow\infty. $$

Proof

By virtue of Lemma 2.2, we can prove the conclusion in a similar way to [16, Lemma 3.8] and [17, Corollary 8.18]. □

Lemma 2.6

Assume that \((V)\) and \((F_{1})\) hold, then there exist constants \(\rho, \alpha>0\) such that \(\Phi|_{\partial B_{\rho}\cap Z_{m}}\geq\alpha\).

Proof

From (2.1) and \((F_{1})\), for all \(u\in E_{\lambda}\) we have
$$\begin{aligned} \Phi_{\lambda}(u) =&\frac{1}{2} \int_{\mathbb{R}^{3}}\bigl( \vert \triangle u \vert ^{2}+a \vert \nabla u \vert ^{2}+\lambda V(x)u^{2}\bigr)\,dx + \frac{b}{4} \biggl( \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \biggr)^{2} \\ &{}+\frac{1}{2} \int_{\mathbb{R}^{3}}u^{2} \vert \nabla u \vert ^{2}\,dx- \int_{\mathbb{R}^{3}}F(x,u)\,dx \\ \geq&\frac{1}{2} \Vert u \Vert ^{2}_{\lambda}- \biggl(\frac{\lambda V_{0}+C_{0}}{2} \Vert u \Vert ^{2}_{2}+ \frac{C_{0}}{p} \Vert u \Vert ^{p}_{p} \biggr). \end{aligned}$$
(2.17)
By virtue of Lemma 2.5, we can choose an integer \(m\geq 1\), for all \(u\in Z_{m}\), satisfying
$$\begin{aligned} & \Vert u \Vert ^{2}_{2}\leq\frac{1}{2(\lambda V_{0}+C_{0})} \Vert u \Vert ^{2}_{\lambda}, \\ & \Vert u \Vert ^{p}_{p}\leq\frac{p}{4C_{0}} \Vert u \Vert ^{p}_{\lambda}. \end{aligned}$$
Combining this with (2.17), one has
$$\Phi_{\lambda}(u)\geq\frac{1}{4} \Vert u \Vert ^{2}_{\lambda}\bigl(1- \Vert u \Vert ^{p-2}_{\lambda} \bigr). $$
Note that, if we let \(\rho= \Vert u \Vert _{\lambda}>0\) be sufficiently small, then \(\Phi_{\lambda}(u)\geq\frac{1}{8}\rho^{2}>0\). □

Lemma 2.7

Assume that \((V)\), \((F_{1})\) and \((F_{2})\) hold, then, for any finite dimensional subspace \(E\subset E_{\lambda}\), there exists \(R=R(E)>0\) such that \(\Phi_{\lambda}|_{E\backslash B_{\rho}}<0\).

Proof

According to the proof of Lemma 2.4, we know that, for any \(u\in E\backslash \{0\}\), there exists a unique \(T=\widetilde{t}(u)>0\) such that
$$v(x)=T^{-1}u(Tx)\in H^{2}\bigl(\mathbb{R}^{3} \bigr) \quad \mbox{and}\quad \Vert v \Vert _{H^{2}}=1. $$
Hence
$$u(x)=Tv\bigl(T^{-1}x\bigr) \quad \mbox{with } \Vert v \Vert _{H^{2}}=1\mbox{ and } T>0. $$
By the equivalence of norms in the finite dimensional space E, there exists \(C_{3}>0\) such that
$$\min\{a,1\} \Vert u \Vert ^{2}_{H^{2}}\leq \Vert u \Vert ^{2}_{\lambda}\leq C_{3} \Vert u \Vert ^{2}_{2}. $$
Combining this with
$$T=\widetilde{t}(u)\rightarrow\infty\quad \mbox{as } \Vert u \Vert _{\lambda}\rightarrow\infty\mbox{ uniformly in }E, $$
we find that, for any \(\delta>0\), there exists a large \(R=R(E,\delta)>0\) such that
$$T=\widetilde{t}(u)\geq\delta\quad \mbox{for all }u\in E\mbox{ with } \Vert u \Vert _{\lambda}\geq R. $$
By \((F_{1})\), there exists \(C_{4}>0\), for all \(x\in\mathbb{R}^{N}\), \(\vert u \vert \leq R_{0}\) such that
$$\bigl\vert F(x,u) \bigr\vert \leq C_{4}u^{2}, $$
where \(R_{0}\) is given by (2.15). Combining (2.1) with \(\Vert v \Vert _{H^{2}}=1\), it follows that for all \(u\in E\backslash\{0\}\)
$$\begin{aligned} \Phi_{\lambda}(u) =&\frac{1}{2} \Vert u \Vert ^{2}_{\lambda}+\frac{b}{4} \Vert \nabla u \Vert ^{4}_{2}+\frac{1}{2} \int_{\mathbb{R}^{3}}u^{2} \vert \nabla u \vert ^{2}\,dx- \int_{\mathbb{R}^{3}}\biggl[\frac{\lambda V_{0}}{2}u^{2}+F(x,u)\biggr] \,dx \\ \leq&\frac{C_{3}}{2} \Vert u \Vert ^{2}_{2}+ \frac{b}{4} \Vert \nabla u \Vert ^{4}_{2}+ \frac{1}{2} \int_{\mathbb{R}^{3}}u^{2} \vert \nabla u \vert ^{2}\,dx- \int_{\mathbb{R}^{3}}\biggl[\frac{\lambda V_{0}}{2}u^{2}+F(x,u)\biggr] \,dx \\ =&\frac{C_{3}-\lambda V_{0}}{2}T^{5} \Vert v \Vert ^{2}_{2}+ \frac{bT^{6}}{4} \Vert \nabla v \Vert ^{4}_{2}-T^{3} \int_{\mathbb{R}^{3}}F(Tx,Tv)\,dx+\frac{1}{2} \int_{\mathbb{R}^{3}}u^{2} \vert \nabla u \vert ^{2}\,dx \\ \leq& T^{6} \biggl(\frac{b}{4}+\frac{C_{3}+\lambda V_{0}+2C_{4}}{2T}- \int_{ \vert Tv \vert \geq R_{0}}\frac{F(Tx,Tv)}{T^{3}}\,dx \biggr)+\frac{1}{2} \int_{\mathbb{R}^{3}}u^{2} \vert \nabla u \vert ^{2}\,dx \\ =&\Psi(T). \end{aligned}$$
(2.18)
Note that \(v\neq0\), then it follows from \((F_{2})\) that
$$\frac{F(Tx,Tv)}{ \vert Tv \vert ^{3}}\rightarrow+\infty\quad \mbox{as }T\rightarrow+\infty. $$
Thus
$$\int_{ \vert Tv \vert \geq R_{0}}\frac{F(Tx,Tv)}{T^{3}}\rightarrow+\infty\quad \mbox{as }T \rightarrow+\infty. $$
Combining this with (2.18), we obtain
$$\Psi(T)\rightarrow-\infty\quad \mbox{as }T\rightarrow+\infty. $$
Thus, there exists a large \(T_{0}>0\) such that
$$\Psi(T)\leq-1 $$
for all \(T\geq T_{0}\). Taking \(\delta=T_{0}\), then there exists a large \(R=R(E)>0\) such that
$$T=\widetilde{t}(u)\geq T_{0} $$
for all \(u\in E\) with \(\Vert u \Vert _{\lambda}\geq R\).

Hence, \(\Phi_{\lambda}(u)<0\) for all \(u\in E\) with \(\Vert u \Vert _{\lambda}\geq R\). □

Proof of Theorem 1.1

Let \(X=E_{\lambda}\), \(Y=Y_{m}\) and \(Z=\overline{Z_{m}}\). Clearly, \(\Phi(0)=0\) and \(\Phi(u)=\Phi(-u)\) due to \((F_{4})\). By virtue of Lemma 2.4, Lemma 2.6, Lemma 2.7 and the fountain theorem (Theorem 3.6 [16]), problem (1.1) possesses infinitely many high energy solutions. □

Proof of Corollary 1.2

Let us consider the Hilbert space
$$H= \biggl\{ u\in H^{1}\bigl(\mathbb{R}^{3}\bigr): \int_{\mathbb{R}^{3}}\bigl(a \vert \nabla u \vert ^{2}+ \lambda\widetilde{V}(x)u^{2}\bigr)\,dx< \infty \biggr\} $$
endowed with the norm
$$\Vert u \Vert = \biggl( \int_{\mathbb{R}^{3}}\bigl(a \vert \nabla u \vert ^{2}+ \lambda\widetilde{V}(x)u^{2}\bigr)\,dx \biggr)^{\frac{1}{2}}. $$
Let
$$\Phi(u)=\frac{1}{2} \int_{\mathbb{R}^{3}}\bigl(a \vert \nabla u \vert ^{2}+ \lambda V(x)u^{2}\bigr)\,dx+\frac{b}{4} \biggl( \int_{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx \biggr)^{2} - \int_{\mathbb{R}^{3}}F(x,u)\,dx,\quad \forall u\in H. $$
Obviously, Φ is a well-defined class \(C^{1}\) functional, and the embedding \(H\hookrightarrow L^{s}\) is compact for \(2\leq s<6\) (see the proof of Lemma 2.2). By Lemma 2.4, Lemma 2.6, Lemma 2.7 and the fountain theorem (Theorem 3.6 [16]), problem (1.3) possesses infinitely many high energy solutions. □

Remark 2.8

In the next paper, we wish to consider the sign-changing solutions for the biharmonic problem like in [19, 21] and so on.

3 Conclusions

In this paper, we consider a sequence of high energy weak solutions for the modified quasilinear fourth-order elliptic equation (1.1) under rather weak conditions. We first prove that the energy functional satisfies the Cerami condition in the well-defined Hilbert space and then prove that the fountain theorem holds under the given conditions by a new technique. Our results extend and improve some recent results.

Declarations

Acknowledgements

This research was supported by the National Science Foundation of China grant 11471187 and 11571197.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors wrote, read, and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

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 Mathematical Sciences, Qufu Normal University, Shandong, P.R. China

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