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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.

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 [211] 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 \(u=+\infty \).

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. 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,

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

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 [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 \(u_{3}\neq u_{1}\), \(u_{3}\neq u_{2}\). □

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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.

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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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Yuanfang, R., Yukun, A. Existence and multiplicity of solutions for nonlocal fourth-order elliptic equations with combined nonlinearities. Bound Value Probl 2020, 130 (2020). https://doi.org/10.1186/s13661-020-01430-4

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Keywords

  • Fourth-order elliptic equation
  • Nonlocal
  • Asymptotically linear
  • Mountain pass theorem
  • Critical point