# Existence and multiplicity of solutions for nonlocal fourth-order elliptic equations with combined nonlinearities

## Abstract

This paper is concerned with the following nonlocal fourth-order elliptic problem:

\begin{aligned} \textstyle\begin{cases} \Delta ^{2}u-m(\int _{\varOmega } \vert \nabla u \vert ^{2} \,dx)\Delta u=a(x) \vert u \vert ^{s-2}u+f(x,u), \quad x\in \varOmega , \\ u=\Delta u=0,\quad x\in \partial \varOmega , \end{cases}\displaystyle \end{aligned}

by using the mountain pass theorem, the least action principle, and the Ekeland variational principle, the existence and multiplicity results are obtained.

## 1 Introduction

In this paper, we consider the following nonlocal fourth-order elliptic problem:

\begin{aligned} \textstyle\begin{cases} \Delta ^{2}u-m(\int _{\varOmega } \vert \nabla u \vert ^{2} \,dx)\Delta u=a(x) \vert u \vert ^{s-2}u+f(x,u), \quad x\in \varOmega , \\ u=\Delta u=0,\quad x\in \partial \varOmega , \end{cases}\displaystyle \end{aligned}
(1.1)

where $$\varOmega \subset R^{N}$$ ($$N>4$$) is a bounded smooth domain, $$m(\cdot )\in C(R^{+},R^{+})$$, $$a(\cdot )\in C(\overline{\varOmega },R^{+})$$, $$s\in (1,2)$$, and $$f\in C(\overline{\varOmega }\times R,R)$$.

Problem (1.1) is related to the stationary problems associated with

\begin{aligned} \frac{\partial ^{2}u}{\partial t^{2}}+\Delta ^{2} u + \biggl(Q+ \int _{\varOmega } \vert \nabla u \vert ^{2}\,dx \biggr) \Delta u=f(x, u, u_{t}). \end{aligned}

This plate model was proposed by Berger  in 1955, as a simplification of the von Karman plate equation which describes large defection of a plate, where the parameter Q describes in-plane forces applied to the plate and the function f represents transverse loads which may depend on the displacement u and the velocity $$u_{t}$$.

Because of the important background, several researchers have considered problem (1.1) by using variational methods when $$a(x)\equiv 0$$,

\begin{aligned} \textstyle\begin{cases} \Delta ^{2}u-m(\int _{\varOmega } \vert \nabla u \vert ^{2} \,dx)\Delta u= f(x,u), \quad x\in \varOmega , \\ u=\Delta u=0,\quad x\in \partial \varOmega , \end{cases}\displaystyle \end{aligned}

with the function m being bounded or unbounded and f having superlinear growth. We refer the readers to  and the references therein.

Recently, in , Ru et al. considered problem (1.1) with $$m(t)=a+bt$$ and a more general f such as

\begin{aligned} \textstyle\begin{cases} \Delta ^{2}u-(a+b\int _{\varOmega } \vert \nabla u \vert ^{2} \,dx)\triangle u =f(x,u, \nabla u,\triangle u), \quad x\in \varOmega , \\ u=\Delta u=0,\quad x\in \partial \varOmega . \end{cases}\displaystyle \end{aligned}

By using an iterative method based on the mountain pass lemma and truncation method developed by De Figueiredo et al. , they proved that the above problem has at least one nontrivial solution.

One of the important conditions in their work is that $$f(x,t)$$ satisfies the famous Ambrosetti–Rabinowitz type condition, for short, which is called the (AR) condition:

(AR condition):

there exist $$\varTheta >2$$ and $$t_{1}>0$$, such that

$$0< \varTheta F(x,t,\xi _{1},\xi _{2})\leq tf(x,t,\xi _{1},\xi _{2}), \quad \forall \vert t \vert \geq t_{1},x\in \varOmega ,(\xi _{1},\xi _{2})\in R^{N+1},$$

where $$F(x,t,\xi _{1},\xi _{2})=\int _{0}^{t} f(x,s,\xi _{1},\xi _{2})\,ds$$.

It is well known that (AR) is a important technical condition to apply the mountain pass theorem. This condition implies that

$$\lim_{u\rightarrow \infty }\frac{F(x,u)}{u^{2}}=\infty .$$

If $$f(x,u)$$ is asymptotically linear at $$u=0$$ or $$u=+\infty$$. then $$f(x,u)$$ does not satisfy the (AR) condition. In , A. Bensedik and M. Bouchekif considered second-order elliptic equations of Kirchhoff type with an asymptotically linear potential

\begin{aligned} \textstyle\begin{cases} -m(\int _{\varOmega } \vert \nabla u \vert ^{2} \,dx)\Delta u=f(x,u), \quad x\in \varOmega , \\ u=0,\quad x\in \partial \varOmega . \end{cases}\displaystyle \end{aligned}

On the other hand, the classical equation involving a biharmonic operator

\begin{aligned} \textstyle\begin{cases} \Delta ^{2}u+c\Delta u=a(x) \vert u \vert ^{s-2}u+f(x,u), \quad x\in \varOmega , \\ u(x)=\Delta u(x)=0,\quad x\in \partial \varOmega , \end{cases}\displaystyle \end{aligned}
(1.2)

has been extensively studied using the mountain pass theorem when $$a(x)\equiv 0$$ and $$f(x,u)$$ is asymptotically linear at $$u=0$$ or $$u=+\infty$$. We refer the reader to [15, 16]. In particular, in , Pu et al. considered problem (1.2) when $$a(x)\neq 0$$.

Until now, there are few works on problem (1.1) when $$a(x)\neq 0$$ and $$f(x,u)$$ does not satisfy the (AR) condition. Inspired by these references, in this paper, we discuss the existence and multiplicity of solutions of problem (1.1) when $$a(x)\neq 0$$ and the nonlinearity f is asymptotically linear at $$u=0$$ or $$u=+\infty$$.

## 2 Preliminaries

Assume that the function $$m(t)$$ satisfies the following conditions:

$$(M)$$:

$$m: R^{+}\rightarrow R^{+}$$ is continuous, nondecreasing, and there exists $$m_{1} \geq m_{0}>0$$ such that

$$m_{0}=\min_{t\in R^{+}}m(t)=m(0),\quad \quad m_{1}=\sup_{t\in R^{+}}m(t).$$

### Remark

In  and , the function $$m(t)$$ is assumed that satisfy $$(M)$$ and there exits $$t_{0} > 0$$ such that $$m (t) = m_{1}$$, $$\forall t > t_{0}$$.

First, we study the nonlinear eigenvalue problem

\begin{aligned} \textstyle\begin{cases} \Delta ^{2}u-m(\int _{\varOmega } \vert \nabla u \vert ^{2} \,dx)\Delta u=\varLambda u, \quad x\in \varOmega , \\ u=0,\quad \quad \Delta u=0,\quad x\in \partial \varOmega . \end{cases}\displaystyle \end{aligned}

Let $$(\lambda _{k},\phi _{k})$$ be the eigenvalue and the corresponding eigenfunction of $$(-\Delta ,H_{0}^{1}(\varOmega ))$$, namely

\begin{aligned} \textstyle\begin{cases} -\Delta \phi _{k}=\lambda _{k} \phi _{k}, \quad x\in \varOmega , \\ \phi _{k}(x)=0, \quad x\in \partial \varOmega . \end{cases}\displaystyle \end{aligned}

Set

$$Lu=\Delta ^{2} u-m\biggl( \int _{\varOmega } \vert \nabla u \vert ^{2}\,dx\biggr) \Delta u.$$

Via some simple computations, we get

\begin{aligned} L\phi _{k} =&\Delta ^{2} \phi _{k}-m\biggl( \int _{\varOmega } \vert \nabla \phi _{k} \vert ^{2}\,dx\biggr) \Delta \phi _{k} \\ =&\biggl[\lambda _{k}^{2}+\lambda _{k}m\biggl( \int _{\varOmega } \vert \nabla \phi _{k} \vert ^{2}\,dx\biggr)\biggr] \phi _{k} \\ =&\biggl[\lambda _{k}^{2}+\lambda _{k}m\biggl( \lambda _{k} \int _{\varOmega } \vert \phi _{k} \vert ^{2} \,dx\biggr)\biggr] \phi _{k}. \end{aligned}

Set

\begin{aligned} \varLambda _{k}= \textstyle\begin{cases} \lambda _{k}^{2}+\lambda _{k}m(\int _{\varOmega } \vert \nabla \phi _{k} \vert ^{2}\,dx), \quad \text{or} \\ \lambda _{k}^{2}+\lambda _{k}m(\lambda _{k}\int _{\varOmega } \vert \phi _{k} \vert ^{2}\,dx) \end{cases}\displaystyle \end{aligned}
(2.1)

and so $$\varLambda _{k}$$ ($$k=1,2,\ldots$$) are the eigenvalues of the operator L associated to the eigenfunction $$\phi _{k}$$.

Assume that the eigenfunctions $$\phi _{k}$$ are suitably normalized with respect to the $$L^{2}(\varOmega )$$ inner product, namely

$$(\phi _{i},\phi _{j})_{L^{2}(\varOmega )}= \textstyle\begin{cases} 0,&i\neq j; \\ 1 ,&i=j. \end{cases}$$

Expression (2.1) can be rewritten as

\begin{aligned} \varLambda _{k}=\lambda _{k}^{2}+\lambda _{k}m \biggl(\lambda _{k} \int _{ \varOmega } \vert \phi _{k} \vert ^{2} \,dx \biggr)=\lambda _{k}^{2}+\lambda _{k}m( \lambda _{k}). \end{aligned}

For each eigenvalue $$\lambda _{k}$$ being repeated as often as multiplicity, recall that

$$0< \lambda _{1}\leq \lambda _{2}\leq \lambda _{3} \leq \cdots \leq \lambda _{k}\rightarrow +\infty ,$$

and if $$(M)$$ holds, then

$$0< \varLambda _{1}\leq \varLambda _{2}\leq \varLambda _{3}\leq \cdots \leq \varLambda _{k}\rightarrow +\infty .$$

Denote

\begin{aligned} \bar{\varLambda }_{k}=\lambda _{k}^{2}+m_{1} \lambda _{k}, \quad k=1,2,\ldots, \end{aligned}

then we know that

\begin{aligned} \varLambda _{k}\leq \bar{\varLambda }_{k}, \quad k=1,2, \ldots. \end{aligned}

It is well known that

$$\lambda _{1}=\inf \biggl\{ \int _{\varOmega } \vert \nabla u \vert ^{2} \,dx: u\in { \mathbf{H}}^{1}_{0}( \varOmega ), \int _{\varOmega } \vert u \vert ^{2} \,dx=1 \biggr\} .$$

Similarly, we have

### Lemma 2.1

Assume that$$(M)$$holds, then

\begin{aligned} \begin{aligned} \varLambda _{1}={}&\inf \biggl\{ \int _{\varOmega } \vert \Delta u \vert ^{2}\,dx+m \biggl( \int _{\varOmega } \vert \nabla u \vert ^{2} \,dx \biggr) \int _{\varOmega } \vert \nabla u \vert ^{2} \,dx: \\ &u \in {\mathbf{H}}^{2}(\varOmega )\cap {\mathbf{H}}^{1}_{0}( \varOmega ), \int _{\varOmega } \vert u \vert ^{2} \,dx=1 \biggr\} . \end{aligned} \end{aligned}

### Proof

Denote

\begin{aligned} \begin{gathered} \inf \biggl\{ \int _{\varOmega } \vert \Delta u \vert ^{2}\,dx+m \biggl( \int _{\varOmega } \vert \nabla u \vert ^{2} \,dx \biggr) \int _{\varOmega } \vert \nabla u \vert ^{2} \,dx: \\ \quad u\in { \mathbf{H}}^{2}( \varOmega )\cap {\mathbf{H}}^{1}_{0}( \varOmega ), \int _{\varOmega } \vert u \vert ^{2} \,dx=1 \biggr\} = \varLambda _{0}, \end{gathered} \end{aligned}

then it is clear that

\begin{aligned} \varLambda _{1} = \lambda _{1}^{2}+\lambda _{1}m(\lambda _{1}) \geq \varLambda _{0}. \end{aligned}

Let $$u_{0}\in {\mathbf{H}}^{2}(\varOmega )\cap {\mathbf{H}}^{1}_{0}(\varOmega )$$ achieve $$\varLambda _{0}$$, then $$\int _{\varOmega } \vert u_{0} \vert ^{2} \,dx=1$$, $$\int _{\varOmega } \vert \nabla u_{0} \vert ^{2} \,dx \geq \lambda _{1}$$ and $$u_{0}=0$$ on ∂Ω, therefore

\begin{aligned} \int _{\varOmega } \vert \nabla u_{0} \vert ^{2} \,dx=- \int _{\varOmega }u_{0}\Delta u_{0} \,dx, \end{aligned}

which implies that

\begin{aligned} \biggl( \int _{\varOmega } \vert \nabla u_{0} \vert ^{2} \,dx \biggr)^{2}= \biggl(- \int _{ \varOmega }u_{0}\Delta u_{0} \biggr)^{2}\,dx \leq \int _{\varOmega } \vert u_{0} \vert ^{2}\,dx \int _{\varOmega } \vert \Delta u_{0} \vert ^{2}\,dx= \int _{\varOmega } \vert \Delta u_{0} \vert ^{2}\,dx, \end{aligned}

then

\begin{aligned} \varLambda _{0} =& \int _{\varOmega } \vert \Delta u_{0} \vert ^{2}\,dx+m \biggl( \int _{\varOmega } \vert \nabla u_{0} \vert ^{2} \,dx \biggr) \int _{\varOmega } \vert \nabla u_{0} \vert ^{2} \,dx \\ \geq & \biggl( \int _{\varOmega } \vert \nabla u_{0} \vert ^{2}\,dx \biggr)^{2}+m \biggl( \int _{\varOmega } \vert \nabla u_{0} \vert ^{2} \,dx \biggr) \int _{\varOmega } \vert \nabla u_{0} \vert ^{2} \,dx \\ \geq & \lambda _{1}^{2}+\lambda _{1}m(\lambda _{1})= \varLambda _{1}. \end{aligned}

So $$\varLambda _{0} = \varLambda _{1}$$.

Let $${\mathbf{H}}={\mathbf{H}}^{2}(\varOmega )\cap {\mathbf{H}}^{1}_{0}( \varOmega )$$ be the Hilbert space equipped with the standard inner product

$$(u,v)_{H}= \int _{\varOmega } (\Delta u \Delta v+ \nabla u\nabla v)\,dx$$

and the deduced norm

$$\Vert u \Vert _{H} ^{2}= \int _{\varOmega } \vert \Delta u \vert ^{2}\,dx+ \int _{\varOmega } \vert \nabla u \vert ^{2}\,dx.$$

It is well know that $$\Vert u \Vert _{H}$$ is equivalent to $$(\int _{\varOmega } \vert \Delta u \vert ^{2} \,dx)^{\frac{1}{2}}$$. And there exists $$\tau >0$$ such that

\begin{aligned} \int _{\varOmega } \vert \Delta u \vert ^{2} \,dx\leq \Vert u \Vert _{H}^{2}\leq \tau \int _{\varOmega } \vert \Delta u \vert ^{2} \,dx. \end{aligned}

Denote

$$\Vert u \Vert ^{2}= \int _{\varOmega } \vert \Delta u \vert ^{2} \,dx+m_{1} \int _{\varOmega } \vert \nabla u \vert ^{2}\,dx$$

and

$$\Vert u \Vert _{m_{0}}^{2}= \int _{\varOmega } \vert \Delta u \vert ^{2} \,dx+m_{0} \int _{\varOmega } \vert \nabla u \vert ^{2}\,dx.$$

It is obvious that the norms $$\Vert u \Vert$$ and $$\Vert u \Vert _{m_{0}}$$ are equivalent to the norm $$\Vert u \Vert _{H}$$ in H. And since $$m_{0}< m_{1}$$, we have

\begin{aligned} \Vert u \Vert ^{2}\geq \Vert u \Vert _{m_{0}}^{2}\geq \theta \Vert u \Vert ^{2}, \end{aligned}

where $$\theta =\frac{m_{0}}{m_{1}}\in (0.1)$$.

Throughout this paper, we denote by C universal positive constants, unless otherwise specified, and

\begin{aligned}& \Vert u \Vert _{\infty }= \Vert u \Vert _{L^{\infty }} \quad \text{for } u\in {\mathbf{L}}^{\infty }(\varOmega ) \text{ or } u\in { \mathbf{C}}(\overline{\varOmega }), \\& \Vert u \Vert _{q}= \biggl( \int _{\varOmega } \vert \nabla u \vert ^{q} \,dx \biggr)^{\frac{1}{q}} \quad \text{for } u\in {\mathbf{L}}^{q}, 1\leq q < +\infty . \end{aligned}

By the Sobolev embedding theorem, there is a positive $$K_{q}$$ such that

\begin{aligned} \Vert u \Vert _{q} \leq K_{q} \Vert u \Vert \quad \text{for } u\in {\mathbf{H}} \text{ and } 1\leq q < \frac{2N}{N-4}. \end{aligned}
(2.2)

Specially, when condition $$(M)$$ holds and $$q=2$$, by Lemma 2.1, then

\begin{aligned} \Vert u \Vert _{2}^{2} \leq \frac{1}{\varLambda _{1}} \Vert u \Vert ^{2}. \end{aligned}
(2.3)

The mountain pass theorem and the Ekeland variational principle are our main tools, which can be found in . □

### Lemma 2.2

LetEbe a real Banach space, and$$I\in C^{1}(E, R)$$satisfy (PS) condition. Suppose

1. 1

There exist$$\rho >0$$, $$\alpha >0$$such that

$$I| _{\partial B_{\rho }}\geq I(0)+\alpha ,$$

where$$B_{\rho }=\{ u\in E| \Vert u \Vert \leq \rho \}$$.

2. 2

There is an$$e\in E$$with$$\Vert e \Vert >\rho$$such that

$$I(e)\leq I(0).$$

Then$$I(u)$$has a critical valuecwhich can be characterized as

$$c=\inf_{\gamma \in \varGamma }\max_{u\in \gamma ([0,1])}I(u),$$

where$$\varGamma =\{\gamma \in C([0,1],E)| \gamma (0)=0,\gamma (1)=e\}$$.

### Lemma 2.3

LetVbe a complete metric space and$$I: V\rightarrow R\cup \{+\infty \}$$be lower semicontinuous, bounded from below. Let$$\varepsilon >0$$be given and$$v\in V$$be such that

$$I(v)\leq \inf_{V}I+\varepsilon .$$

Then there exists$$u\in V$$such that

$$I(u)\leq I(v), \quad\quad d(v,u)\leq 1$$

and for all$$w\neq u$$in V,

$$I(w)> F(u)-\varepsilon d(v,w).$$

## 3 Main results

A function $$u\in \mathbf{H}$$ is called a weak solution of (1.1) if

$$\int _{\varOmega }\Delta u \Delta v\,dx+m \biggl( \int _{\varOmega } \vert \nabla u \vert ^{2}\,dx \biggr) \int _{\varOmega }\nabla u\nabla v\,dx - \int _{\varOmega }a(x) \vert u \vert ^{s-2}uv \,dx= \int _{\varOmega }f(x,u)v\,dx$$

holds for any $$v\in \mathbf{H}$$. Let $$J:\mathbf{H}\rightarrow R$$ be the functional defined by

$$J(u)=\frac{1}{2} \int _{\varOmega } \vert \Delta u \vert ^{2}\,dx+ \frac{1}{2}{M} \biggl( \int _{\varOmega } \vert \nabla u \vert ^{2}\,dx \biggr) -\frac{1}{s} \int _{\varOmega }a(x) \vert u \vert ^{s}\,dx- \int _{\varOmega }F(x,u)\,dx,$$

where

$$M(t)= \int _{0}^{t}m(s)\,ds, \quad\quad F(t)= \int _{0}^{t}f(x,s)\,ds.$$

It is easy to see that $$J\in C^{1}({\mathbf{H}}, R)$$ and the critical points of J in H correspond to the weak solutions of problem (1.1).

We make the following assumptions.

$$(A)$$:

$$a(x)\in {\mathbf{C}}(\overline{\varOmega })$$, $$a(x)\geq 0$$, $$\forall x\in \overline{\varOmega }$$ and $$\Vert a(x) \Vert _{\infty }=\bar{a}>0$$;

$$(F_{0})$$:

$$tf(x,t)\geq 0$$ for $$x\in \overline{{\varOmega }}$$, $$t\in {\mathbf{R}}$$;

$$(F_{1})$$:

$$\lim_{ \vert t \vert \rightarrow 0}\frac{f(x,t)}{t}=p(x)$$ uniformly a.e. $$x\in {\varOmega }$$, where $$0< p(x)\in L^{\infty }(\varOmega )$$, and $$\Vert p(x) \Vert _{\infty }<\theta \varLambda _{1}$$;

$$(F_{2})$$:

$$\lim_{ \vert t \vert \rightarrow +\infty }\frac{f(x,t)}{t}=l$$ ($$-\infty < l< + \infty$$) uniformly a.e. $$x\in {\varOmega }$$.

Our first main result is concluded as the following theorem:

### Theorem 3.1

Assume the function$$m(t)$$satisfies$$(M)$$, $$a(x)$$satisfies$$(A)$$, and the nonlinearity$$f(x,t)$$satisfies$$(F_{1})$$and$$(F_{2})$$, then problem (1.1) has at least one solution if$$l< \varLambda _{1}$$.

### Proof

It is easy to see, from condition $$(F_{1})$$, that $$f(x,0)= 0$$ for $$x\in \varOmega$$. So $$u=0$$ is the trivial solution of (1.1). From condition $$(F_{2})$$, we can take $$\varepsilon =\frac{1}{2}(\varLambda _{1}-l)>0$$, and there exists $$T>0$$ such that

$$f(x,t)t\leq (l+\varepsilon )t^{2}$$

for all $$\vert t \vert \geq T$$ and a.e. $$x\in \varOmega$$. By the continuity of F, there exists $$C>0$$ such that

\begin{aligned} \bigl\vert F(x,t) \bigr\vert \leq{} \frac{l+\varepsilon }{2}t^{2}+C \end{aligned}

for all $$(x,t)\in \varOmega \times R$$. On the other hand, from $$(M)$$ it follows that

\begin{aligned} m_{0} t \leq {M}(t)= \int _{0}^{t} m(s)\,ds\leq m_{1} t, \quad \text{for }t>0. \end{aligned}
(3.1)

Then we have

\begin{aligned} J(u) =&\frac{1}{2} \int _{\varOmega } \vert \Delta u \vert ^{2}\,dx+ \frac{1}{2}{M} \biggl( \int _{\varOmega } \vert \nabla u \vert ^{2} \,dx \biggr)-\frac{1}{s} \int _{0}^{t}a(x) \vert u \vert ^{s} \,dx- \int _{\varOmega }F(x,u)\,dx \\ \geq &\frac{1}{2} \int _{\varOmega } \vert \Delta u \vert ^{2}\,dx+ \frac{1}{2}m_{0} \int _{\varOmega } \vert \nabla u \vert ^{2} \,dx- \frac{1}{s}\bar{a} \int _{\varOmega } \vert u \vert ^{s}\,dx \\ & {} -\frac{l+\varepsilon }{2} \int _{\varOmega } \vert u \vert ^{2}\,dx-C \vert \varOmega \vert \\ \geq &\frac{1}{2} \Vert u \Vert ^{2}-\frac{1}{s}K_{s} \bar{a} \Vert u \Vert ^{s}- \frac{l+\varepsilon }{2\varLambda _{1}} \Vert u \Vert ^{2}-C \vert \varOmega \vert \\ =&\frac{\varLambda _{1}-l-\varepsilon }{2\varLambda _{1}} \Vert u \Vert ^{2}- \frac{1}{s}K_{s} \bar{a} \Vert u \Vert ^{s}-C \vert \varOmega \vert , \end{aligned}

which shows that J is coercive. Moreover, conditions $$(F_{1})$$ and $$(F_{2})$$ imply that J is weakly lower semicontinuous in H. Therefore we get a global minimum $$u_{1}$$ of J.

Next, we prove $$u_{1}\neq 0$$, so it is a nontrivial solution of (1.1). From condition $$(F_{1})$$, there exists $$C>0$$ such that

$$\bigl\vert f(x,t) \bigr\vert \leq C \vert t \vert ,$$

for all $$\vert t \vert$$ small enough and $$x\in \varOmega$$. It follows that

$$\bigl\vert F(x,t) \bigr\vert \leq \frac{C}{2}t^{2},$$

for all $$\vert t \vert$$ small enough and $$x\in \varOmega$$. From condition $$(A)$$, we can chose $$v\in {\mathbf{H}}$$ such that

$$\int _{\varOmega }a(x) \vert v \vert ^{s}\,dx>0.$$

Then we have

\begin{aligned}& \limsup_{t\rightarrow 0}\frac{J(tv)}{t^{s}} \\& \quad =\limsup_{t\rightarrow 0} \frac{\frac{1}{2}\int _{\varOmega } \vert \Delta (tv) \vert ^{2}\,dx+ \frac{1}{2}{M}(\int _{\varOmega } \vert \nabla (tv) \vert ^{2} \,dx)-\frac{1}{s}\int _{\varOmega }a(x) \vert tv \vert ^{s}\,dx-\int _{\varOmega }F(x,tv)\,dx}{t^{s}} \\& \quad \leq \limsup_{t\rightarrow 0} \frac{\frac{1}{2}\int _{\varOmega } \vert \Delta (tv) \vert ^{2}\,dx+ \frac{1}{2}m_{1}(\int _{\varOmega } \vert \nabla (tv) \vert ^{2} \,dx)-\frac{1}{s}\int _{\varOmega }a(x) \vert tv \vert ^{s}\,dx-\int _{\varOmega }F(x,tv)\,dx}{t^{s}} \\& \quad \leq \limsup_{t\rightarrow 0}\biggl(\frac{t^{2-s}}{2} \Vert v \Vert ^{2}- \frac{1}{s} \int _{\varOmega }a(x) \vert v \vert ^{s}\,dx+ \frac{Ct^{2-s}}{2} \int _{\varOmega }v^{2}\,dx\biggr) \\& \quad < 0. \end{aligned}

Therefore, we get that $$J(u_{1})<0$$. It is clear that $$J(0)=0$$. Thus, $$u_{1}$$ is a nontrivial solution of (1.1). □

Our second result is the following theorem:

### Theorem 3.2

Assume the function$$m(t)$$satisfies$$(M)$$, $$a(x)$$satisfies$$(A)$$, and the nonlinearity$$f(x,t)$$satisfies$$(F_{0})$$, $$(F_{1})$$, and$$(F_{2})$$, then there exists a positive constant$$a_{0}$$such that problem (1.1) has at least three nontrivial solutions if$$\bar{a}< a_{0}$$and$$\bar{\varLambda }_{1}< l<+\infty$$.

Before proving Theorem 3.2, we give two lemmas.

### Lemma 3.1

Suppose the conditions of Theorem 3.2hold, then there exists a positive constant$$a_{0}$$such thatJsatisfies the following conditions for$$\bar{a}< a_{0}$$and$$\bar{\varLambda }_{1}< l<+\infty$$:

1. 1.

There exist constants$$\rho >0$$, $$\alpha >0$$such that$$J| _{\partial B_{\rho }}\geq \alpha$$with$$B_{\rho }=\{ u\in {\mathbf{H}} : \Vert u \Vert \leq \rho \}$$;

2. 2.

$$J(t\varphi _{1})\rightarrow -\infty$$as$$t\rightarrow +\infty$$.

### Proof

(Claim 1) By $$(F_{1})$$ and $$(F_{2})$$, there exists $$C>0$$ such that for all $$(x,t)\in \varOmega \times R$$ and $$p\in (1,\frac{N+4}{N-4})$$, we have

$$F(x,t)\leq \frac{1}{4}\bigl( \bigl\Vert p(x) \bigr\Vert _{\infty }+\theta \varLambda _{1}\bigr)t^{2}+C \vert t \vert ^{p+1}.$$

From inequalities (2.2), (2.3) and (3.1), we have

\begin{aligned} J(u) =&\frac{1}{2} \int _{\varOmega } \vert \Delta u \vert ^{2}\,dx+ \frac{1}{2}m \biggl( \int _{\varOmega } \vert \nabla u \vert ^{2} \,dx \biggr)-\frac{1}{s} \int _{0}^{t}a(x) \vert u \vert ^{s} \,dx- \int _{\varOmega }F(x,u)\,dx \\ \geq &\frac{1}{2} \int _{\varOmega } \vert \Delta u \vert ^{2}\,dx+ \frac{1}{2}m_{0} \int _{\varOmega } \vert \nabla u \vert ^{2} \,dx- \frac{1}{s}\bar{a} \int _{\varOmega } \vert u \vert ^{s}\,dx \\ & {} -\frac{1}{4}\bigl( \bigl\Vert p(x) \bigr\Vert _{\infty }+ \theta \varLambda _{1}\bigr) \Vert u \Vert ^{2}_{2}-C \Vert u \Vert ^{p+1}_{p+1} \\ \geq & \frac{\theta }{2} \Vert u \Vert ^{2}-\frac{1}{s} \bar{a}K_{s} \Vert u \Vert ^{s}- \frac{1}{4} \frac{( \Vert p(x) \Vert _{\infty }+\theta \varLambda _{1})}{\varLambda _{1}} \Vert u \Vert ^{2} -CK_{p+1} \Vert u \Vert ^{p+1} \\ =& \biggl(\frac{\theta \varLambda _{1}- \Vert p(x) \Vert _{\infty }}{4\varLambda _{1}}- \frac{1}{s}\bar{a}K_{s} \Vert u \Vert ^{s-2} -CK_{p+1} \Vert u \Vert ^{p-1} \biggr) \Vert u \Vert ^{2}. \end{aligned}

Setting

$$a_{0}=\frac{s}{2K_{s} K_{p+1}^{\frac{2-s}{p-1}}} \biggl( \frac{\theta \varLambda _{1}- \Vert p(x) \Vert _{\infty }}{8\varLambda _{1}} \biggr)^{ \frac{p-s+1}{p-1}},\quad \quad \rho = \biggl( \frac{\theta \varLambda _{1}- \Vert p(x) \Vert _{\infty }}{8\varLambda _{1}CK_{p+1}} \biggr)^{\frac{1}{p-1}},$$

when $$\bar{a}\leq a_{0}$$ and $$\Vert u \Vert =\rho$$, it follows that

$$J(u)\geq \biggl( \frac{\theta \varLambda _{1}- \Vert p(x) \Vert _{\infty }}{16\varLambda _{1}} \biggr) \Vert \rho \Vert ^{2}= \alpha >0.$$

So, Claim 1 is proved.

(Claim 2) By $$(F_{2})$$ and for $$l>\bar{\varLambda }_{1}$$, there exists $$C>0$$ such that

$$F(x,t)\geq \frac{1}{4}(l+ \bar{\varLambda }_{1})t^{2}-C$$

for all $$(x,t)\in \varOmega \times R$$. Let $$\lambda _{1}$$ and $$\phi _{1}$$ be the first eigenvalue and eigenfunction of $$(-\Delta ,H_{0}^{1}(\varOmega ))$$ with $$\int _{\varOmega } \vert \phi _{1} \vert ^{2} \,dx=1$$. We know that

$$\bar{ \varLambda }_{1} = \int _{\varOmega } \vert \Delta \phi _{1} \vert ^{2}\,dx+m_{1} \int _{\varOmega } \vert \nabla \phi _{1} \vert ^{2} \,dx= \lambda _{1}^{2}+m_{1} \lambda _{1}.$$

Then, we have

\begin{aligned} J(t\phi _{1}) =& \frac{1}{2} \int _{\varOmega } \bigl\vert \Delta (t\phi _{1}) \bigr\vert ^{2}\,dx+ \frac{1}{2}m\biggl( \int _{\varOmega } \bigl\vert \nabla (t\phi _{1}) \bigr\vert ^{2} \,dx\biggr) \\ & {} -\frac{1}{s} \int _{\varOmega }a(x) \vert t\phi _{1} \vert ^{s}\,dx- \int _{\varOmega }F(x,t \phi _{1})\,dx \\ \leq &\frac{t^{2}}{2} \int _{\varOmega } \vert \Delta \phi _{1} \vert ^{2}\,dx+ \frac{t^{2}}{2} m_{1} \int _{\varOmega } \vert \nabla \phi _{1} \vert ^{2} \,dx \\ & {} -\frac{t^{s}}{s} \int _{\varOmega }a(x) \vert \phi _{1} \vert ^{s}\,dx-\frac{t^{2}}{4}(l+ \bar{\varLambda }_{1}) \int _{\varOmega } \vert \phi _{1} \vert ^{2} \,dx+C \vert \varOmega \vert \\ =& \frac{t^{2}}{4}(\bar{\varLambda }_{1}-l)-\frac{t^{s}}{s} \int _{\varOmega }a(x) \vert \phi _{1} \vert ^{s}\,dx+C \vert \varOmega \vert . \end{aligned}

Hence, $$J(t\psi _{1}) \rightarrow -\infty$$, $$t\rightarrow +\infty$$.

The proof of Lemma 3.1 is completed. □

Let

\begin{aligned} f^{+}(x,t)= \textstyle\begin{cases} f(x,t), &t \geq 0, \\ 0, & t < 0, \end{cases}\displaystyle \end{aligned}

and

\begin{aligned} f^{-}(x,t)= \textstyle\begin{cases} f(x,t), &t \leq 0, \\ 0, &t > 0. \end{cases}\displaystyle \end{aligned}

Define functionals $$J^{\pm }: \mathbf{H} \rightarrow \mathbf{R}$$ as follows:

$$J^{\pm }(u)=\frac{1}{2} \int _{\varOmega } \vert \Delta u \vert ^{2}\,dx+ \frac{1}{2}{m} \biggl( \int _{\varOmega } \vert \nabla u \vert ^{2}\,dx \biggr) -\frac{1}{s} \int _{\varOmega }a(x) \vert u \vert ^{s}\,dx- \int _{\varOmega }F^{\pm }(x,u)\,dx,$$

where $$F^{\pm }(t)=\int _{0}^{t}f^{\pm }(x,s)\,ds$$.

### Lemma 3.2

Assume that$$(M)$$, $$(A)$$and$$(F_{0})$$$$(F_{2})$$hold, and$$\bar{\varLambda }_{1} < l < +\infty$$, then$$J^{\pm }(u)$$satisfies the (PS) condition.

### Proof

We just prove that $$J^{+}(u)$$ satisfies the (PS) condition. The proof for $$J^{-}(u)$$ is similar. Let $$\{u_{n}\}\in \mathbf{H}$$ be a (PS) sequence, namely

\begin{aligned}& J^{+}(u_{n})\rightarrow c, \end{aligned}
(3.2)
\begin{aligned}& \nabla J^{+}(u_{n})\rightarrow 0. \end{aligned}
(3.3)

Firstly, we claim that $$\{u_{n}\}$$ is bounded in H. If not, we may assume that $$\Vert u_{n} \Vert \rightarrow +\infty$$ as $$n\rightarrow +\infty$$. Let $$w_{n}=\frac{u_{n}}{ \Vert u_{n} \Vert }$$, then $$\Vert w_{n} \Vert =1$$. Passing to a subsequence, we may assume that there exists $$w\in {\mathbf{H}}$$ such that

\begin{aligned} \textstyle\begin{cases} w_{n}\rightharpoonup w\quad \text{in } {\mathbf{H}}, \\ w_{n}\rightarrow w\quad \text{in } {\mathbf{L}^{r}}(\varOmega ), 1\leq r \leq \frac{2N}{N-4}, \\ w_{n}\rightarrow w\quad \text{a.e. in }\varOmega . \end{cases}\displaystyle \end{aligned}
(3.4)

By $$(F_{1})$$ and $$(F_{2})$$, we see that there exist $$C_{1}$$ and $$C_{2}$$ such that

\begin{aligned} \biggl\vert \frac{f(x,t)}{t} \biggr\vert \leq C_{1}, \quad\quad \biggl\vert \frac{F(x,t)}{t^{2}} \biggr\vert \leq C_{2} \end{aligned}
(3.5)

for all $$(x,t)\in \varOmega \times {\mathbf{R}}$$ and define

\begin{aligned} \frac{f(x,t)}{t}\bigg| _{t=0}=\lim_{t\rightarrow 0} \frac{f(x,t)}{t}, \quad\quad \frac{F(x,t)}{t^{2}}\bigg| _{t=0}=\lim _{t\rightarrow 0} \frac{F(x,t)}{t^{2}}. \end{aligned}

Then we claim that $$w\neq 0$$. Otherwise, if $$w\equiv 0$$, we know that $$w_{n}\rightarrow 0$$ strongly in $${\mathbf{L}}^{r}(\varOmega )$$. Dividing (3.2) by $$\Vert u_{n} \Vert ^{2}$$, we have

\begin{aligned} \frac{J^{+}(u_{n})}{ \Vert u_{n} \Vert ^{2}} =& \frac{1}{2 \Vert u_{n} \Vert ^{2}} \biggl( \int _{\varOmega } \vert \Delta u_{n} \vert ^{2}\,dx+m \biggl( \int _{\varOmega } \vert \nabla u_{n} \vert ^{2} \,dx \biggr) \biggr) \\ & {} - \frac{1}{s \Vert u_{n} \Vert ^{2-s}} \int _{\varOmega }a(x) \bigl\vert w_{n}(x) \bigr\vert ^{s}\,dx- \int _{ \varOmega } \frac{F^{+}(x,u_{n})}{ \Vert u_{n} \Vert ^{2}}\,dx \\ =& o(1). \end{aligned}

It follows from (3.1) and (3.5) that

\begin{aligned} \frac{\theta }{2} \leq & \frac{1}{2 \Vert u_{n} \Vert ^{2}} \biggl( \int _{\varOmega } \vert \Delta u_{n} \vert ^{2}\,dx+m_{0} \int _{\varOmega } \vert \nabla u_{n} \vert ^{2} \,dx \biggr) \\ \leq & \frac{1}{2 \Vert u_{n} \Vert ^{2}} \biggl( \int _{\varOmega } \vert \Delta u_{n} \vert ^{2}\,dx+ m \biggl( \int _{\varOmega } \vert \nabla u_{n} \vert ^{2} \,dx \biggr) \biggr) \\ =& \frac{1}{s \Vert u_{n} \Vert ^{2-s}} \int _{\varOmega }a(x) \bigl\vert w_{n}(x) \bigr\vert ^{s}\,dx+ \int _{ \varOmega } \frac{F^{+}(x,u_{n})}{ \Vert u_{n} \Vert ^{2}}\,dx+o(1) \\ \leq & \frac{\bar{a}}{s \Vert u_{n} \Vert ^{2-s}} \int _{\varOmega } \bigl\vert w_{n}(x) \bigr\vert ^{s}\,dx+C_{2} \int _{\varOmega } \bigl\vert w_{n}(x) \bigr\vert ^{2}\,dx+o(1)\rightarrow 0, \end{aligned}

which is impossible, so $$w\neq 0$$.

Let us define

\begin{aligned} \varOmega _{0}=\bigl\{ x\in \varOmega| w(x)=0\bigr\} , \quad\quad \varOmega _{1}=\bigl\{ x\in \varOmega| w(x)\neq 0\bigr\} . \end{aligned}

Then, for all $$v\in {\mathbf{H}}$$, we have

\begin{aligned} \biggl\vert \int _{\varOmega _{0}} \frac{f^{+}(x,u_{n})}{u_{n}}w_{n}v\,dx \biggr\vert \leq & C_{1} \int _{\varOmega _{0}} \vert w_{n} \vert \vert v \vert \,dx \\ \leq & C_{1} \biggl( \int _{\varOmega _{0}} \vert w_{n} \vert ^{2}\,dx \biggr)^{ \frac{1}{2}} \biggl( \int _{\varOmega _{0}} \vert v \vert ^{2}\,dx \biggr)^{\frac{1}{2}}. \end{aligned}

So,

\begin{aligned} \lim_{n\rightarrow +\infty } \int _{\varOmega _{0}} \frac{f^{+}(x,u_{n})}{u_{n}}w_{n}v\,dx=0= \int _{\varOmega _{0}} l w^{+} v \,dx, \end{aligned}
(3.6)

where $$w^{+}(x)=\max {\{w(x), 0\}}$$. On the other hand, since $$\Vert u_{n} \Vert \rightarrow +\infty$$, we have $$\vert u_{n}(x) \vert = \Vert u_{n} \Vert \vert w_{n}(x) \vert \rightarrow +\infty$$ for $$x\in \varOmega _{1}$$. Therefore, by $$(F_{2})$$ and the dominated convergence theorem, we get

\begin{aligned} \lim_{n\rightarrow +\infty } \int _{\varOmega _{1}} \frac{f^{+}(x,u_{n})}{u_{n}}w_{n}v\,dx= \int _{\varOmega _{1}}\lim_{n \rightarrow +\infty } \frac{f^{+}(x,u_{n})}{u_{n}}w_{n}v \,dx= \int _{ \varOmega _{1}} l w^{+} v \,dx. \end{aligned}
(3.7)

Combining (3.6) and (3.7), we obtain

\begin{aligned} \lim_{n\rightarrow +\infty } \int _{\varOmega } \frac{f^{+}(x,u_{n})}{u_{n}}w_{n}v\,dx= \int _{\varOmega } l w^{+} v \,dx. \end{aligned}
(3.8)

Now, (3.3) implies that, for all $$v\in {\mathbf{H}}$$, we have

\begin{aligned} \bigl(\nabla J^{+}(u_{n}),v\bigr) =& \int _{\varOmega }\Delta u_{n}\Delta v\,dx+m \biggl( \int _{\varOmega } \vert \nabla u_{n} \vert ^{2} \,dx \biggr) \int _{\varOmega }\nabla u_{n} \nabla v \,dx \\ & {} - \int _{\varOmega }a(x) \bigl\vert u_{n}(x) \bigr\vert ^{s-1}v \,dx- \int _{\varOmega } f^{+}(x,u_{n})v \,dx\rightarrow 0. \end{aligned}

Dividing by $$\Vert u_{n} \Vert$$, we get

\begin{aligned} \begin{aligned}[b] & \int _{\varOmega }\Delta w_{n}\Delta v\,dx+m \biggl( \int _{\varOmega } \vert \nabla u_{n} \vert ^{2} \,dx \biggr) \int _{\varOmega }\nabla w_{n}\nabla v \,dx \\ &\quad{} -\frac{1}{ \Vert u_{n} \Vert ^{2-s}} \int _{\varOmega }a(x) \bigl\vert w_{n}(x) \bigr\vert ^{s-1}v \,dx- \int _{ \varOmega } \frac{f^{+}(x,u_{n})}{u_{n}}w_{n}v \,dx\rightarrow 0. \end{aligned} \end{aligned}
(3.9)

Since

$$\Vert u_{n} \Vert ^{2}= \int _{\varOmega } \vert \Delta u_{n} \vert ^{2}\,dx+m_{1} \int _{ \varOmega } \vert \nabla u_{n} \vert ^{2}\,dx\rightarrow +\infty$$

as $$n\rightarrow +\infty$$, we can suppose that there exists a subsequence, still denoted $$\{\int _{\varOmega } \vert \nabla u_{n} \vert ^{2}\,dx\}$$, such that

\begin{aligned} \int _{\varOmega } \vert \nabla u_{n} \vert ^{2}\,dx\rightarrow +\infty , \quad n\rightarrow +\infty , \end{aligned}
(3.10)

otherwise, there exists $$K>0$$ such that

\begin{aligned} \int _{\varOmega } \vert \nabla u_{n} \vert ^{2}\,dx\leq K, \end{aligned}

and furthermore, there exist a subsequence, still denoted $$\{\int _{\varOmega } \vert \nabla u_{n} \vert ^{2}\,dx\}$$, and a constant $$t'\geq 0$$ such that

\begin{aligned} \int _{\varOmega } \vert \nabla u_{n} \vert ^{2}\,dx\rightarrow t', \quad n\rightarrow + \infty. \end{aligned}
(3.11)

In case (3.10) holds, by $$(M)$$, we have

\begin{aligned} \lim_{n\rightarrow +\infty }m \biggl( \int _{\varOmega } \vert \nabla u_{n} \vert ^{2} \,dx \biggr)=m_{1}. \end{aligned}
(3.12)

Combining (3.4), (3.8), (3.9) and (3.10), as $$n\rightarrow +\infty$$, we obtain

\begin{aligned} \int _{\varOmega }\Delta w\Delta v\,dx+m_{1} \int _{\varOmega }\nabla w\nabla v \,dx= \int _{\varOmega } lw^{+}v \,dx, \quad \forall v\in { \mathbf{H}}. \end{aligned}
(3.13)

Taking $$v=\phi _{1}$$ in (3.13), we have

\begin{aligned} \int _{\varOmega }\Delta w\Delta \phi _{1} \,dx+m_{1} \int _{\varOmega }\nabla w \nabla \phi _{1} \,dx= \int _{\varOmega } lw^{+}\phi _{1} \,dx. \end{aligned}
(3.14)

Noticing that $$\phi _{1}$$ is the positive solution of

\begin{aligned} \textstyle\begin{cases} \Delta ^{2}u+m_{1}\Delta u=\bar{\varLambda }_{1} u, \quad \text{in }\varOmega , \\ u=0,\qquad \Delta u=0,\quad \text{on }\partial \varOmega , \end{cases}\displaystyle \end{aligned}

we have

\begin{aligned} \int _{\varOmega }\Delta w\Delta \phi _{1} \,dx+m_{1} \int _{\varOmega }\nabla w \nabla \phi _{1} \,dx= \int _{\varOmega } \bar{\varLambda }_{1}w\phi _{1} \,dx. \end{aligned}
(3.15)

Thus, from (3.14) and (3.15), we get

\begin{aligned} \int _{\varOmega } lw^{+}\phi _{1} \,dx= \int _{\varOmega } \bar{\varLambda }_{1}w \phi _{1} \,dx. \end{aligned}
(3.16)

If $$w(x)\geq 0$$ a.e. in Ω, since $$w(x)\neq 0$$, we have $$\int _{\varOmega }w\phi _{1} \,dx>0$$. Then (3.15) implies that

\begin{aligned} \int _{\varOmega } lw\phi _{1} \,dx= \int _{\varOmega } lw^{+}\phi _{1} \,dx= \int _{ \varOmega } \bar{\varLambda }_{1}w\phi _{1} \,dx, \end{aligned}

which contradicts $$l>\bar{\varLambda }_{1}$$. Otherwise, let $$\varOmega _{-}=\{x\in \varOmega| w(x)<0\}$$ and suppose $$\vert \varOmega _{-} \vert >0$$. Then $$\int _{\varOmega _{-}}-w\phi _{1} \,dx>0$$ and $$\int _{\varOmega }w^{+}\phi _{1} \,dx>\int _{\varOmega }w\phi _{1} \,dx>0$$. It follows from (3.15) again that

\begin{aligned} \int _{\varOmega } lw^{+}\phi _{1} \,dx= \int _{\varOmega } \bar{\varLambda }_{1}w \phi _{1} \,dx< \int _{\varOmega } \bar{\varLambda }_{1}w^{+}\phi _{1} \,dx, \end{aligned}

which contradicts $$l>\bar{\varLambda }_{1}$$.

So $$\{u_{n}\}$$ is bounded in X.

In case (3.11) holds, by $$(M)$$, we have

\begin{aligned} \lim_{n\rightarrow +\infty }m \biggl( \int _{\varOmega } \vert \nabla u_{n} \vert ^{2} \,dx \biggr)=m\bigl(t'\bigr)=m'\leq m_{1}. \end{aligned}
(3.17)

Combining (3.4), (3.8), (3.9) and (3.17), as $$n\rightarrow +\infty$$, we obtain

\begin{aligned} \int _{\varOmega }\Delta w\Delta v\,dx+m' \int _{\varOmega }\nabla w\nabla v \,dx= \int _{\varOmega } lw^{+}v \,dx, \quad \forall v\in { \mathbf{H}}. \end{aligned}
(3.18)

Taking $$v=\phi _{1}$$ in (3.18), we have

\begin{aligned} \int _{\varOmega }\Delta w\Delta \phi _{1} \,dx+m' \int _{\varOmega }\nabla w \nabla \phi _{1} \,dx= \int _{\varOmega } lw^{+}\phi _{1} \,dx. \end{aligned}
(3.19)

Notice that $$\phi _{1}$$ is also the positive solution of

\begin{aligned} \textstyle\begin{cases} \Delta ^{2}u+m'\Delta u=\varLambda '_{1} u, \quad \text{in }\varOmega , \\ u=0,\qquad \Delta u=0,\quad \text{on }\partial \varOmega, \end{cases}\displaystyle \end{aligned}

where $$\varLambda '_{1}=\lambda _{1}^{2}+m'\lambda _{1}$$. Then we have

\begin{aligned} \int _{\varOmega }\Delta w\Delta \phi _{1} \,dx+m' \int _{\varOmega }\nabla w \nabla \phi _{1} \,dx= \int _{\varOmega } \varLambda '_{1}w\phi _{1} \,dx. \end{aligned}
(3.20)

From (3.19) and (3.20), we get

\begin{aligned} \int _{\varOmega } lw^{+}\phi _{1} \,dx= \int _{\varOmega } \varLambda '_{1}w\phi _{1} \,dx. \end{aligned}
(3.21)

Notice that for $$\varLambda '_{1}\leq \bar{\varLambda }_{1}$$, similar to the discussions in case (3.10) holds, (3.21) implies a contradiction to $$l>\bar{\varLambda }_{1}$$.

So $$\{u_{n}\}$$ is bounded in X.

Now, since Ω is bounded and $$(F_{1})$$, $$(F_{2})$$ hold, by using the Sobolev embedding theorem and the standard procedures, we can easily prove that $$\{u_{n}\}$$ has a convergent subsequence. The proof of the lemma is completed. □

### Proof of Theorem 3.2.

From the proof of Lemma 3.1, it is easy to see that $$J^{+}(u)$$ and $$J^{-}(u)$$ satisfy the conditions of Lemma 3.1. So there exist $$\rho >0$$, $$\alpha >0$$, and $$e\in {\mathbf{H}}$$ with $$\Vert e \Vert >\rho$$ such that

\begin{aligned} J^{\pm }(u)\big| _{\partial B_{\rho }}\geq \alpha >0, \quad\quad J^{\pm }(e)< 0. \end{aligned}

It is clear that $$J^{\pm }(0)=0$$. Moreover, by Lemma 3.2, the functionals $$J^{\pm }$$ satisfy the (PS) condition. By Lemma 2.2, we know that $$J^{\pm }$$ has the critical value $$c^{\pm }$$, respectively, which can be characterized as

$$c^{\pm }=\inf_{\gamma \in \varGamma }\max_{u\in \gamma ([0,1])}J^{\pm }(u),$$

where $$\varGamma =\{\gamma \in C([0,1],{\mathbf{H}})| \gamma (0)=0,\gamma (1)=e \}$$. So there exist critical points $$u_{1}, u_{2} \in {\mathbf{H}}$$ such that

\begin{aligned} J^{+}(u_{1})=c^{+}>0, \quad\quad J^{-}(u_{2})=c^{-}>0. \end{aligned}

Since $$f^{+}(x,t)\geq 0$$ and $$f^{-}(x,t)\leq 0$$, by the comparison principles for some fourth order elliptic problems , $$u_{1}$$ is a positive solution of (1.1) and $$u_{2}$$ is a negative solution of (1.1).

Next, we prove that problem (1.1) has another solution $$u_{3} \in {\mathbf{H}}$$ such that $$J(u_{3})<0$$. For $$\rho >0$$ given by Lemma 3.1, define $$B_{\rho }=\{u\in E: \Vert u \Vert \leq \rho \}$$ and then $$B_{\rho }$$ is a complete metric space with the distance $$\operatorname{dist}(u,v)= \Vert u-v \Vert$$ for $$u,v\in B_{\rho }$$. By Lemma 3.1, we know that

\begin{aligned} J(u)| _{\partial B_{\rho }}\geq \alpha >0. \end{aligned}
(3.22)

Clearly, $$J\in C^{1}(B_{\rho },R)$$, so J is bounded from below on $$B_{\rho }$$. And we know that J is lower semicontinuous.

Similar to the proof of Theorem 3.1, there exists $$v \in {\mathbf{H}}$$ such that

\begin{aligned}& \lim_{t\rightarrow 0}\frac{J(tv)}{t^{p}}< 0. \end{aligned}

Then letting $$c_{1}=\inf \{J(u):u\in B_{\rho }\}$$, we get that $$c_{1}<0$$. By Lemma 2.3, for any $$k>0$$, there is a $$\{u_{k}\}$$ such that

$$c_{1}\leq J(u_{k})\leq c_{1}+\frac{1}{k}.$$

Now we claim that $$\Vert u_{k} \Vert <\rho$$ for k large enough. Otherwise, if $$\Vert u_{k} \Vert =\rho$$ for infinitely many k, and, without loss of generality, we may suppose that $$\Vert u_{k} \Vert =\rho$$ for all $$k>1$$. It follows from (3.22) that $$J(u_{k})\geq \alpha >0$$. Letting $$k\rightarrow \infty$$, we see that $$0>c_{1}\geq \alpha >0$$, which is a contradiction.

For any $$u\in E$$ with $$\Vert u \Vert =1$$, let

$$w_{k}=u_{k}+tu$$

for any fixed $$k\geq 1$$. We get

$$\Vert w_{k} \Vert \leq \Vert u_{k} \Vert +t,$$

so $$w_{k}\in B_{\rho }$$ for $$t>0$$ small enough. It follows from Lemma 2.3 that

$$J(w_{k})=J(u_{k}+tu)\geq J(u_{k})- \frac{t}{k} \Vert u \Vert .$$

Thus, we have

$$J'(u_{k})=\lim_{t\rightarrow 0^{+}}\frac{J(u_{k}+tu)-J(u_{k})}{t} \geq -\frac{1}{k}$$

and

$$J'(u_{k})=\lim_{t\rightarrow 0^{+}}\frac{J(u_{k}-tu)-J(u_{k})}{t} \leq \frac{1}{k}.$$

Then $$\vert J'(u_{k}) \vert \leq \frac{1}{k}\rightarrow 0$$ and $$J(u_{k})\rightarrow c_{1}$$ as $$k\rightarrow \infty$$. Therefore $$\{u_{k}\}$$ is a (PS) sequence at level $$c_{1}$$. From Lemma 3.2, $$\{u_{k}\}$$ has a convergent subsequence. Hence, we see that there exists $$u_{3}\in {\mathbf{H}}$$ such that $$J'(u_{3})=0$$ and $$J(u_{3})=c_{1}<0$$. Thus, $$u_{3}$$ is a nontrivial weak solution of (1.1) and $$u_{3}\neq u_{1}$$, $$u_{3}\neq u_{2}$$. □

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### Acknowledgements

We would like to thank the referee for his/her valuable comments and helpful suggestions which have led to an improvement of the presentation of this paper.

Not applicable.

## Funding

This work was supported by the Fundamental Research Funds for the Central Universities (2632020PY02) and the National Natural Foundation of China-NSAF (Grant No. 11571092).

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

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Correspondence to Ru Yuanfang.

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