High energy solutions of modified quasilinear fourth-order elliptic equation

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.

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.