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 [1] 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 [2–11] and the references therein.

Recently, in [12], 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. [13], 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 [14], 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 [17], 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 [14] and [18], 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 [19]. □

Lemma 2.2

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

• 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 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. 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. $J(t\varphi _{1})\rightarrow -\infty$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 [20], $u_{1}$ 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 $u_{3}\neq u_{1}$, $u_{3}\neq u_{2}$. □