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# High energy solutions of modified quasilinear fourth-order elliptic equation

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## 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.

## 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  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, 58] and the references therein. It is well known that fourth-order elliptic equation has been widely studied since Lazer and Mckenna  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, 1013]. In , 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 , 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 , 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 , Xu and Chen also used condition $$(V')$$ to study the problem (1.3).

More recently, Cheng and Tang  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 $$(f_{7})$$  and $$(f_{4})$$; $$(F_{2})$$ is weaker than $$(g_{2})$$ . Furthermore, we do not require λ large enough, but we only need $$\lambda\geq 1$$. Therefore, our results extend and improve Theorem 1 , Theorem 1.2 , Theorem 1.3 , Theorem 1.1 , Theorem 1.4  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

$$f(x,u)u-4F(x,u)+\alpha u^{2}=\frac{2u^{2}\ln(1+u^{2}-2u^{4}+\alpha u^{6}+2\alpha u^{4}+\alpha u^{2})}{(1+u^{2})^{2}}\geq0.$$

## 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 ), $$\Phi_{\lambda}$$ 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 , 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 ), 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 ), 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.

## 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.

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## Acknowledgements

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

## Author information

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

Correspondence to Aixia Qian.

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### Competing interests

The authors declare that they have no competing interests. 