Nonlinear biharmonic boundary value problem

Abstract

We consider the nonlinear biharmonic equation with variable coefficient and polynomial growth nonlinearity and Dirichlet boundary condition. We get two theorems. One theorem says that there exists at least one bounded solution under some condition. The other one says that there exist at least two solutions, one of which is a bounded solution and the other of which has a large norm under some condition. We obtain this result by the variational method, generalized mountain pass geometry and the critical point theory of the associated functional.

MSC:35J20, 35J25, 35Q72.

1 Introduction

Let Ω be a bounded domain in $R^n$ with smooth boundary ∂ Ω and $L^2(\mathrm{\Omega })$ be a square integrable function space defined on Ω. Let Δ be the elliptic operator and ${\mathrm{\Delta }}^2$ be the biharmonic operator. Let $c\in R$. In this paper we study the following nonlinear biharmonic equation with Dirichlet boundary condition:

$$\begin{array}{r}{\mathrm{\Delta }}^{2}u+c\mathrm{\Delta }u=a(x)g(u)\text{in }\mathrm{\Omega },\\ u=0,\mathrm{\Delta }u=0\text{on }\partial \mathrm{\Omega },\end{array}$$

(1.1)

where $a:\overline{\mathrm{\Omega }}\to R$ is a continuous function which changes sign in Ω.

We assume that g satisfies the following conditions:

(g1) $g\in C(R,R)$,

(g2) there are constants $a_1,a_2\ge 0$ such that

$$|g(u)|\le a_1+a_2{|u|}^{\mu -1},$$

where $2<\mu <\frac{2n}{n-2}$ if $n\ge 3$,

(g3) there exists a constant $r_0\ge 0$ such that

$$0<\mu G(\xi )=\mu {\int }_0^{\xi }g(t)dt\le \xi g(\xi )\text{for }|\xi |\ge r_0,$$

(g4) $g(u)=o(|u|)$ as $u\to 0$.

We note that (g3) implies the existence of the positive constants $a_3$, $a_4$, $a_5$ such that

$$\frac{1}{\mu }(\xi g(\xi )+a_3)\ge G(\xi )+a_4\ge a_5{|\xi |}^{\mu }\text{for }\xi \in R.$$

(1.2)

Remark 1.1 The real number ξ in the definition (g3) is not automatically nonnegative. The reason is as follows.

Since $0<\mu G(\xi )<\xi g(\xi )$ and $\mu >2$, $G(\xi )>0$ and $\xi g(\xi )>0$. By $\xi g(\xi )>0$, we have two cases: one case is that $\xi >0$ and $g(\xi )>0$. The other case is that $\xi <0$ and $g(\xi )<0$. Thus ξ is not nonnegative.

Remark 1.2 We obtain the boundedness of $\frac{1}{2}g(u)u-G(u)$ as follows.

By the condition (g3), $\frac{1}{2}g(u)u\ge \frac{1}{2}\mu G(u)$ for $|u|\ge r_0$. Since $\mu >2$, $\frac{1}{2}-\frac{1}{\mu }>0$, and $G(u)+a_4\ge a_5{|u|}^{\mu }$ in (1.2),

$$\begin{array}{rcl}\frac{1}{2}g(u)u-G(u)& \ge & \frac{1}{2}\mu G(u)-G(u)\\ =& \mu (\frac{1}{2}-\frac{1}{\mu })G(u)\\ \ge & \mu (\frac{1}{2}-\frac{1}{\mu })(a_5{|u|}^{\mu }-a_4).\end{array}$$

Thus we obtain the boundedness of $\frac{1}{2}g(u)u-G(u)$.

Remark 1.3 (i) Assumption (g4) implies that (1.1) has a trivial solution.

• (ii) If $n=1$, (g2) can be dropped. If $n=2$, it suffices that

  $$|g(u)|\le a_{1}expp(\xi ),$$

where $p(\xi ){\xi }^{-2}\to 0$ as $|\xi |\to \mathrm{\infty }$.

• (iii) If $n\ge 3$ and $g(\xi )={{\xi }^{+}}^{1+ϵ}$, where ${\xi }^{+}=max\{\xi ,0\}$ and $ϵ>0$ is a small number, then (g1)-(g4) are satisfied.

The eigenvalue problem

$$\begin{array}{c}\mathrm{\Delta }u+\lambda u=0\text{in }\mathrm{\Omega },\hfill \\ u=0\text{on }\partial \mathrm{\Omega }\hfill \end{array}$$

has infinitely many eigenvalues ${\lambda }_k$, $k\ge 1$, and corresponding eigenfunctions ${\varphi }_k$, $k\ge 1$, suitably normalized with respect to the $L^2(\mathrm{\Omega })$ inner product, where each eigenvalue ${\lambda }_k$ is repeated as often as its multiplicity. The eigenvalue problem

$$\begin{array}{c}{\mathrm{\Delta }}^{2}u+c\mathrm{\Delta }u=\mathrm{\Lambda }u\text{in }\mathrm{\Omega },\hfill \\ u=0,\mathrm{\Delta }u=0\text{on }\partial \mathrm{\Omega }\hfill \end{array}$$

has also infinitely many eigenvalues ${\lambda }_k({\lambda }_k-c)$, $k\ge 1$ and corresponding eigenfunctions ${\varphi }_k$, $k\ge 1$. We note that ${\lambda }_1({\lambda }_1-c)\le {\lambda }_2({\lambda }_2-c)\le \cdots \to +\mathrm{\infty }$, and that ${\varphi }_1(x)>0$ for $x\in \mathrm{\Omega }$.

Khanfir and Lassoued [1] showed the existence of at least one solution for the nonlinear elliptic boundary problem when g is locally Hölder continuous on $R_{+}$. Choi and Jung [2] showed that the problem

$$\begin{array}{r}{\mathrm{\Delta }}^{2}u+c\mathrm{\Delta }u=b{u}^{+}+s\text{in }\mathrm{\Omega },\\ u=0,\mathrm{\Delta }u=0\text{on }\partial \mathrm{\Omega }\end{array}$$

(1.3)

has at least two nontrivial solutions when ($c<{\lambda }_1$, ${\lambda }_1({\lambda }_1-c)<b<{\lambda }_2({\lambda }_2-c)$ and $s<0$) or (${\lambda }_1<c<{\lambda }_2$, $b<{\lambda }_1({\lambda }_1-c)$ and $s>0$). The authors obtained these results by using the variational reduction method. The authors [3] also proved that when $c<{\lambda }_1$, ${\lambda }_1({\lambda }_1-c)<b<{\lambda }_2({\lambda }_2-c)$ and $s<0$, (1.3) has at least three nontrivial solutions by using degree theory. Tarantello [4] also studied

$$\begin{array}{r}{\mathrm{\Delta }}^{2}u+c\mathrm{\Delta }u=b({(u+1)}^{+}-1),\\ u=0,\mathrm{\Delta }u=0\text{on }\partial \mathrm{\Omega }.\end{array}$$

(1.4)

She showed that if $c<{\lambda }_1$ and $b\ge {\lambda }_1({\lambda }_1-c)$, then (1.4) has a negative solution. She obtained this result by degree theory. Micheletti and Pistoia [5] also proved that if $c<{\lambda }_1$ and $b\ge {\lambda }_2({\lambda }_2-c)$ then (1.4) has at least four solutions by the variational linking theorem and Leray-Schauder degree theory.

In this paper we are trying to find the weak solutions of (1.1), that is,

$${\int }_{\mathrm{\Omega }}({\mathrm{\Delta }}^{2}u+c\mathrm{\Delta }u-a(x)g(u))vdx=0\text{for any }v\in H,$$

where the space H is introduced in Section 2. Let us set

$${\mathrm{\Omega }}^{+}=\{x\in \mathrm{\Omega }\mid a(x)>0\},{\mathrm{\Omega }}^{-}=\{x\in \mathrm{\Omega }\mid a(x)<0\}$$

and let

$$a^{+}=a\cdot {\chi }_{{\mathrm{\Omega }}^{+}},a^{-}=-a\cdot {\chi }_{{\mathrm{\Omega }}^{-}}.$$

Since $a(x)$ changes sign, the open subsets ${\mathrm{\Omega }}^{+}$ and ${\mathrm{\Omega }}^{-}$ are nonempty. Now we can write $a=a^{+}-a^{-}$. Our main results are as follows.

Theorem 1.1 Assume that ${\lambda }_k<c<{\lambda }_{k+1}$, g satisfies (g1)-(g4) and $g(u)u-\mu G(u)$ is bounded. Then (1.1) has at least one bounded nontrivial solution.

Theorem 1.2 Assume that ${\lambda }_k<c<{\lambda }_{k+1}$, g satisfies (g1)-(g4), $g(u)u-\mu G(u)$ is not bounded and there exists a small $ϵ>0$ such that ${\int }_{{\mathrm{\Omega }}^{-}}{a}^{-}(x)dx<ϵ$. Then (1.1) has at least two solutions, (i) one of which is nontrivial and bounded, and (ii) the other of which has a large norm such that

$$\underset{x\in \mathrm{\Omega }}{max}|u(x)|>M\mathit{\text{for some }}M.$$

The outline of Theorem 1.1 and Theorem 1.2 is as follows: In Section 2, we prove that the corresponding functional $I(u)$ of (1.1), which is introduced in (2.1), is continuous and Fréchet differentiable and satisfies the $(\mathit{PS})$ condition. In Section 3, we prove Theorem 1.1. In Section 4, we prove Theorem 1.2 by the variational method, the generalized mountain pass geometry and the critical point theory.

2 Palais-Smale condition

Any element u in $L^2(\mathrm{\Omega })$ can be written as

$$u=\sum h_k{\varphi }_k\text{with }\sum h_k^2<\mathrm{\infty }.$$

We define a subspace H of $L^2(\mathrm{\Omega })$ as follows:

$$H=\{u\in L^2(\mathrm{\Omega })|\sum |{\lambda }_k({\lambda }_k-c)|h_k^2<\mathrm{\infty }\}.$$

Then this is a Banach space with a norm

$$\parallel u\parallel ={[\sum |{\lambda }_k({\lambda }_k-c)\left|h_k^2\right]}^{\frac{1}{2}}.$$

Since ${\lambda }_k\to +\mathrm{\infty }$ and c is fixed, we have

• (i) ${\mathrm{\Delta }}^{2}u+c\mathrm{\Delta }u\in H$ implies $u\in H$,

• (ii) $\parallel u\parallel \ge C{\parallel u\parallel }_{L^2(\mathrm{\Omega })}$, for some $C>0$,

• (iii) ${\parallel u\parallel }_{L^2(\mathrm{\Omega })}=0$ if and only if $\parallel u\parallel =0$,

which are proved in [6].

Let

$$\begin{array}{c}{H}_{+}=\{u\in H\mid h_k=0\text{ if }{\lambda }_k({\lambda }_k-c)<0\},\hfill \\ H_{-}=\{u\in H\mid h_k=0\text{ if }{\lambda }_k({\lambda }_k-c)>0\}.\hfill \end{array}$$

Then $H=H_{-}\oplus H_{+}$. Let $P_{+}$ be the orthogonal projection on $H_{+}$ and $P_{-}$ be the orthogonal projection on $H_{-}$.

We are looking for the weak solutions of (1.1). The weak solutions of (1.1) coincide with the critical points of the associated functional

$$\begin{array}{c}I(u)\in C^1(H,R),\hfill \\ I(u)={\int }_{\mathrm{\Omega }}[\frac{1}{2}{|\mathrm{\Delta }u|}^2-\frac{c}{2}{|\mathrm{\nabla }u|}^2-{\int }_{\mathrm{\Omega }}a(x)G(u)]dx\hfill \\ \phantom{I(u)}=\frac{1}{2}({\parallel P_{+}u\parallel }^2-{\parallel P_{-}u\parallel }^2)-{\int }_{\mathrm{\Omega }}a(x)G(u)dx.\hfill \end{array}$$

(2.1)

By (g1) and (g2), I is well defined. By Proposition 2.1, $I\in C^1(H,R)$ and I is Fréchet differentiable in H.

Proposition 2.1 Assume that ${\lambda }_k<c<{\lambda }_{k+1}$, $k\ge 1$, and that g satisfies (g1)-(g4). Then $I(u)$ is continuous and Fréchet differentiable in H with Fréchet derivative

$$I^{\prime }(u)h={\int }_{\mathrm{\Omega }}[\mathrm{\Delta }u\cdot \mathrm{\Delta }h-c\mathrm{\nabla }u\cdot \mathrm{\nabla }h-a(x)g(u)h]dx.$$

(2.2)

If we set

$$K(u)={\int }_{\mathrm{\Omega }}a(x)G(u)dx,$$

then $K^{\prime }(u)$ is continuous with respect to weak convergence, $K^{\prime }(u)$ is compact, and

$$K^{\prime }(u)h={\int }_{\mathrm{\Omega }}a(x)g(u)hdx\mathit{\text{for all }}h\in H.$$

This implies that $I\in C^1(H,R)$ and $K(u)$ is weakly continuous.

The proof of Proposition 2.1 is the same as that of Appendix B in [7].

Proposition 2.2 (Palais-Smale condition)

Assume that ${\lambda }_k<c<{\lambda }_{k+1}$, $k\ge 1$, and g satisfies (g1)-(g4). We also assume that $g(u)u-\mu G(u)$ is bounded or that there exists an $ϵ>0$ such that ${\int }_{{\mathrm{\Omega }}^{-}}{a}^{-}(x)dx<ϵ$. Then $I(u)$ satisfies the Palais-Smale condition.

Proof Suppose that $(u_m)$ is a sequence with $I(u_m)\le M$ and $I^{\prime }(u_m)\to 0$ as $m\to \mathrm{\infty }$. Then by (g2), (g3), and the Hölder inequality and the Sobolev Embedding Theorem, for large m and $\mu >2$ with $u=u_m$, we have

$$\begin{array}{rcl}M+\frac{1}{2}\parallel u\parallel & \ge & I(u)-\frac{1}{2}{I}^{\prime }(u)u={\int }_{\mathrm{\Omega }}[\frac{1}{2}a(x)g(u)u-a(x)G(u)]dx\\ =& {\int }_{\mathrm{\Omega }}{a}^{+}(x)[\frac{1}{2}g(u)u-G(u)]dx-{\int }_{\mathrm{\Omega }}{a}^{-}(x)[\frac{1}{2}g(u)u-G(u)]dx\\ \ge & (\frac{1}{2}-\frac{1}{\mu })\mu {\int }_{\mathrm{\Omega }}{a}^{+}(x)\cdot G(u)dx-\underset{\mathrm{\Omega }}{max}|\frac{1}{2}g(u)u-G(u)|{\int }_{{\mathrm{\Omega }}^{-}}{a}^{-}(x)dx\\ \ge & (\frac{1}{2}-\frac{1}{\mu })\mu {\int }_{\mathrm{\Omega }}{a}^{+}(x)\cdot (a_5{|u|}^{\mu }-a_4)dx\\ -\underset{\mathrm{\Omega }}{max}|\frac{1}{2}g(u)u-G(u)|{\int }_{{\mathrm{\Omega }}^{-}}{a}^{-}(x)dx.\end{array}$$

Since $\frac{1}{2}g(u)u-G(u)$ is bounded or there exists an $ϵ>0$ such that ${\int }_{{\mathrm{\Omega }}^{-}}{a}^{-}(x)<ϵ$; we have

$$1+\parallel u\parallel \ge M_1{\int }_{\mathrm{\Omega }}{|u|}^{\mu }dx\ge M_2{\left({\int }_{\mathrm{\Omega }}{|u|}^{2}dx\right)}^{\frac{1}{2}\cdot \mu }.$$

(2.3)

Moreover since

$$|I^{\prime }(u_m)\phi |\le \parallel \phi \parallel$$

(2.4)

for large m and all $\phi \in H$, choosing $\phi =P_{+}{u}_m\in H_{+}$ gives

$$\begin{array}{rcl}{\parallel P_{+}{u}_m\parallel }^2& =& {\int }_{\mathrm{\Omega }}({\mathrm{\Delta }}^2{u}_m+c\mathrm{\Delta }{u}_m)\cdot P_{+}{u}_{m}dx\\ =& {\int }_{\mathrm{\Omega }}a(x)g(u_m)P_{+}{u}_{m}dx\\ \le & {\int }_{\mathrm{\Omega }}|a(x)||g(u_m)||u_m|dx\\ \le & {\parallel a\parallel }_{\mathrm{\infty }}{\int }_{\mathrm{\Omega }}(a_1|u_m|+a_2{|u_m|}^{\mu })dx\\ \le & C_1{\int }_{\mathrm{\Omega }}{|u_m|}^{\mu }dx+C_2{\parallel u_m\parallel }_{L^2(\mathrm{\Omega })}\\ \le & C_1{\int }_{\mathrm{\Omega }}{|u_m|}^{\mu }dx+C_2^{\prime }\parallel u_m\parallel .\end{array}$$

Taking $\phi =-P_{-}{u}_m$ in (2.4) yields

$$\begin{array}{rcl}{\parallel P_{-}{u}_m\parallel }^2& =& {\int }_{\mathrm{\Omega }}({\mathrm{\Delta }}^2{u}_m+c\mathrm{\Delta }{u}_m)\cdot (-P_{-}{u}_m)dx\\ =& {\int }_{\mathrm{\Omega }}a(x)g(u_m)\cdot (-P_{-}{u}_m)dx\\ \le & {\int }_{\mathrm{\Omega }}|a(x)||g(u_m)||u_m|dx\\ \le & C_3{\int }_{\mathrm{\Omega }}{|u_m|}^{\mu }dx+C_4\parallel u_m\parallel .\end{array}$$

Thus, by (2.3), we have

$$\begin{array}{rcl}{\parallel u_m\parallel }^2& =& {\parallel P_{+}{u}_m\parallel }^2+{\parallel P_{-}{u}_m\parallel }^2\le M_3{\int }_{\mathrm{\Omega }}{|u_m|}^{\mu }dx+M_4\parallel u_m\parallel \\ \le & M_5(1+\parallel u_m\parallel )+M_4\parallel u_m\parallel \le M_6(1+\parallel u_m\parallel ),\end{array}$$

from which the boundedness of $(u_m)$ follows. Thus $(u_m)$ converges weakly in H. Since $P_{\pm }{I}^{\prime }(u_m)=\pm P_{\pm }{u}_m+P_{\pm }\tilde{\mathcal{P}}(u_m)$ with $\tilde{\mathcal{P}}$ compact and the weak convergence of $P_{\pm }{u}_m$ imply the strong convergence of $P_{\pm }{u}_m$ and hence $(\mathit{PS})$ condition holds. □

3 Proof of Theorem 1.1

We shall show that $I(u)$ satisfies the generalized mountain pass geometrical assumptions.

We recall the generalized mountain pass geometry.

Let $H=V\oplus X$, where $V\ne \{0\}$ and is finite dimensional. Suppose that $I\in C^1(H,R)$, satisfies the Palais-Smale condition, and

• (i) there are constants $\rho ,\alpha >0$ and a bounded neighborhood $B_{\rho }$ of 0 such that $I{|}_{\partial B_{\rho }\cap X}\ge \alpha$, and

• (ii) there is an $e\in \partial B_1\cap X$ and $R>\rho$ such that if $Q=({\overline{B}}_R\cap V)\oplus \{re\mid 0<r<R\}$, then $I{|}_{\partial Q}\le 0$.

Then I possesses a critical value $b\ge \alpha$. Moreover b can be characterized as

$$b=\underset{\gamma \in \mathrm{\Gamma }}{inf}\underset{u\in Q}{max}I(\gamma (u)),$$

where

$$\mathrm{\Gamma }=\{\gamma \in C(\overline{Q},H)\mid \gamma =\mathit{id}\text{ on }\partial Q\}.$$

Let $H_k=span\{{\varphi }_1,\dots ,{\varphi }_k\}$. Then $H_k$ is a subspace of H such that

$$H=\underset{k\in N}{⨁}{H}_k\text{and}H=H_k\oplus H_k^{\perp }.$$

Let

$$B_r=\{u\in H\mid \parallel u\parallel \le r\}.$$

We have the following generalized mountain pass geometrical assumptions.

Lemma 3.1 Assume that ${\lambda }_k<c<{\lambda }_{k+1}$ and g satisfies (g1)-(g4). Then

• (i) there are constants $\rho >0$, $\alpha >0$ and a bounded neighborhood $B_{\rho }$ of 0 such that $I{|}_{\partial B_{\rho }\cap H_k^{\perp }}\ge \alpha$, and

• (ii) there is an $e\in \partial B_1\cap H_k^{\perp }$ and $R>\rho$ such that if $Q=({\overline{B}}_R\cap H_k)\oplus \{re\mid 0<r<R\}$, then $I{|}_{\partial Q}\le 0$, and

• (iii) there exists $u_0\in H$ such that $\parallel u_0\parallel >\rho$ and $I(u_0)\le 0$.

Proof (i) Let $u\in H_k^{\perp }$. Then

$${\int }_{\mathrm{\Omega }}({\mathrm{\Delta }}^{2}u+c\mathrm{\Delta }u)udx\ge {\lambda }_{k+1}({\lambda }_{k+1}-c){\parallel u\parallel }_{L^2(\mathrm{\Omega })}^2>0.$$

Thus by (g2), (g4), and the Hölder inequality, we have

$$\begin{array}{rcl}I(u)& =& \frac{1}{2}{\parallel P_{+}u\parallel }^2-\frac{1}{2}{\parallel P_{-}u\parallel }^2-{\int }_{\mathrm{\Omega }}a(x)G(u)dx\\ \ge & \frac{1}{2}{\parallel P_{+}u\parallel }^2-{\parallel a\parallel }_{\mathrm{\infty }}{\int }_{\mathrm{\Omega }}{C}_1{|u|}^{\mu }dx\\ \ge & \frac{1}{2}{\parallel P_{+}u\parallel }^2-{\parallel a\parallel }_{\mathrm{\infty }}{C}_1^{\prime }{\parallel u\parallel }^{\mu }\end{array}$$

for $C_1,C_1^{\prime }>0$. Since $\mu >2$, there exist $\rho >0$ and $\alpha >0$ such that if $u\in \partial B_{\rho }$, then $I(u)\ge \alpha$.

• (ii) Let $B_r$ be a ball with radius $r>0$, e be a fixed element in $\partial B_1\cap H_k^{\perp }$ and $u\in ({\overline{B}}_r\cap H_k)\oplus \{re\mid 0<r\}$. Then $u=v+w$, $v\in B_r\cap H_k$, $w=re$. We note that

  $$\text{since }v\in H_k,{\int }_{\mathrm{\Omega }}({\mathrm{\Delta }}^{2}v+c\mathrm{\Delta }v)vdx\le {\lambda }_k({\lambda }_k-c){\parallel v\parallel }_{L^2(\mathrm{\Omega })}^2<0.$$

Thus we have

$$\begin{array}{rcl}I(u)& =& \frac{1}{2}{r}^2-\frac{1}{2}{\parallel P_{-}v\parallel }^2-{\int }_{\mathrm{\Omega }}a(x)G(v+re)dx\\ \le & \frac{1}{2}{r}^2+\frac{1}{2}({\lambda }_k({\lambda }_k-c)){\parallel v\parallel }_{L^2(\mathrm{\Omega })}^2-{\int }_{{\mathrm{\Omega }}^{+}}a(x)(a_5|v+re{|}^{\mu }-a_4)dx.\end{array}$$

Since $\mu >2$, there exists $R>0$ such that if $u\in Q=({\overline{B}}_R\cap H_k)\oplus \{re\mid 0<r<R\}$, then $I(u)<0$.

• (iii) If we choose $\psi \in H$ such that $\parallel \psi \parallel =1$, $\psi \ge 0$ in Ω and $supp(\psi )\subset {\mathrm{\Omega }}^{+}$, then we have

  $$\begin{array}{rcl}I(t\psi )& \le & \frac{1}{2}{\parallel P_{+}(t\psi )\parallel }^2-\frac{1}{2}{\parallel P_{-}(t\psi )\parallel }^2-{\int }_{{\mathrm{\Omega }}^{+}}a(x)(a_5{t}^{\mu }{\psi }^{\mu }-a_4)dx\\ \le & \frac{1}{2}{\parallel t\psi \parallel }^2-{\int }_{{\mathrm{\Omega }}^{+}}a(x)(a_5{t}^{\mu }{\psi }^{\mu }-a_4)dx\\ =& \frac{1}{2}{t}^2-{\int }_{{\mathrm{\Omega }}^{+}}a(x)(a_5{t}^{\mu }{\psi }^{\mu }-a_4)dx\end{array}$$

for all $t>0$. Since $\mu >2$, for $t_0$ great enough, $u_0=t_0\psi$ is such that $\parallel u_0\parallel >\rho$ and $I(u_0)\le 0$. □

Proof of Theorem 1.1 By Proposition 2.1 and Proposition 2.2, $I(u)\in C^1(H,R)$ and satisfies the Palais-Smale condition. By Lemma 3.1, there are constants $\rho >0$, $\alpha >0$ and a bounded neighborhood $B_{\rho }$ of 0 such that $I{|}_{\partial B_{\rho }\cap H_m^{\perp }}\ge \alpha$, and there is an $e\in \partial B_1\cap H_k^{\perp }$ and $R>\rho$ such that if $Q=({\overline{B}}_R\cap H_k)\oplus \{re\mid 0<r<R\}$, then $I{|}_{\partial Q}\le 0$, and there exists $u_0\in H$ such that $\parallel u_0\parallel >\rho$ and $I(u_0)\le 0$. By the generalized mountain pass theorem, $I(u)$ has a critical value $b\ge \alpha$. Moreover, b can be characterized as

$$b=\underset{\gamma \in \mathrm{\Gamma }}{inf}\underset{u\in Q}{max}I(\gamma (u)),$$

where

$$\mathrm{\Gamma }=\{\gamma \in C(\overline{Q},H)\mid \gamma =\mathit{id}\text{ on }\partial Q\}.$$

We denote by $\tilde{u}$ a critical point of I such that $I(\tilde{u})=b$. We claim that there exists a constant $C>0$ such that

$${\parallel a^{+}{(x)}^{\frac{1}{\mu }}\tilde{u}\parallel }_{L^2(\mathrm{\Omega })}\le C{(1+L{\int }_{{\mathrm{\Omega }}^{-}}{a}^{-}(x)dx)}^{\frac{1}{\mu }},$$

where $L={max}_{\mathrm{\Omega }}|\frac{1}{2}g(\tilde{u})\tilde{u}-G(\tilde{u})|$.

In fact, we have

$$b\le \underset{0\le t\le 1}{max}I(t{u}_0)$$

and

$$\begin{array}{rcl}I(t{u}_0)& =& t^2(\frac{1}{2}{\parallel P_{+}{u}_0\parallel }^2-\frac{1}{2}{\parallel P_{-}{u}_0\parallel }^2)-{\int }_{\mathrm{\Omega }}a(x)G(t{u}_0)dx\\ \le & t^2{\parallel u_0\parallel }^2-{\int }_{\mathrm{\Omega }}{a}^{+}(x)G(t{u}_0)dx+{\int }_{\mathrm{\Omega }}{a}^{-}(x)G(t{u}_0)dx\\ \le & t^2{\parallel u_0\parallel }^2-a_5{t}^{\mu }{\int }_{\mathrm{\Omega }}{a}^{+}(x)u_0^{\mu }dx+a_4{\int }_{\mathrm{\Omega }}{a}^{+}(x)dx+a_5{t}^{\mu }{\int }_{\mathrm{\Omega }}{a}^{-}(x)u_0^{\mu }dx\\ =& C{t}^2-C{t}^{\mu }+C+C^{\prime }{t}^{\mu }.\end{array}$$

Since $0\le t\le 1$, b is bounded: $b<\tilde{C}$.

We can write

$$\begin{array}{rcl}b& =& I(\tilde{u})-\frac{1}{2}{I}^{\prime }(\tilde{u})\tilde{u}\\ =& {\int }_{\mathrm{\Omega }}a(x)(\frac{1}{2}g(\tilde{u})\tilde{u}-G(\tilde{u}))dx\\ =& {\int }_{\mathrm{\Omega }}{a}^{+}(x)(\frac{1}{2}g(\tilde{u})\tilde{u}-G(\tilde{u}))dx-{\int }_{\mathrm{\Omega }}{a}^{-}(x)(\frac{1}{2}g(\tilde{u})\tilde{u}-G(\tilde{u}))dx\\ \ge & (\frac{1}{2}-\frac{1}{\mu }){\int }_{\mathrm{\Omega }}{a}^{+}(x)g(\tilde{u})\tilde{u}dx-\underset{\mathrm{\Omega }}{max}|\frac{1}{2}g(\tilde{u})\tilde{u}-G(\tilde{u})|{\int }_{{\mathrm{\Omega }}^{-}}{a}^{-}(x)dx\\ \ge & (\frac{1}{2}-\frac{1}{\mu })\mu {\int }_{\mathrm{\Omega }}{a}^{+}(x)(a_3|\tilde{u}{|}^{\mu }-a_4)dx-L{\int }_{{\mathrm{\Omega }}^{-}}{a}^{-}(x)dx,\end{array}$$

where $L={max}_{\mathrm{\Omega }}|\frac{1}{2}g(\tilde{u})\tilde{u}-G(\tilde{u})|$. Thus we have

$$C^{\prime }(1+L{\int }_{{\mathrm{\Omega }}^{-}}{a}^{-}(x)dx)\ge {\int }_{\mathrm{\Omega }}{a}^{+}(x)|\tilde{u}{|}^{\mu }dx\ge C^{″}{\left[{\int }_{\mathrm{\Omega }}{(a^{+}{(x)}^{\frac{1}{\mu }}|\tilde{u}|)}^{2}dx\right]}^{\frac{\mu }{2}}$$

(3.1)

for some constants $C^{\prime },C^{″}>0$, from which we conclude that $\tilde{u}$ is bounded and the proof of Theorem 1.1 is complete. □

4 Proof of Theorem 1.2

Assume that $\frac{1}{2}g(u)u-G(u)$ is not bounded and there exists an $ϵ>0$ such that ${\int }_{{\mathrm{\Omega }}^{-}}{a}^{-}(x,t)dx<ϵ$. By Proposition 2.1 and Proposition 2.2, $I\in C^1(H,R)$ and satisfies the Palais-Smale condition. By Lemma 3.1 and the generalized mountain pass theorem, $I(u)$ has a critical value b with critical point $\tilde{u}$ such that $I(\tilde{u})=b$. If ${\int }_{{\mathrm{\Omega }}^{-}}{a}^{-}(x)dx$ is sufficiently small, by (3.1), we have

$${\int }_{\mathrm{\Omega }}{a}^{+}(x)|\tilde{u}{|}^{\mu }dx\le C$$

for $C>0$, from which we can conclude that $\tilde{u}$ is bounded and the proof of Theorem 1.2(i) is complete.

Next we shall prove Theorem 1.2(ii). We may assume that $R_n<R_{n+1}$ for all $n\in N$. Let us set $D_n=B_{R_n}\cap H_n$, $\partial D_n=\partial B_{R_n}\cap H_n$.

Lemma 4.1 Assume that g satisfies (g1)-(g4), $\frac{1}{2}g(u)u-G(u)$ is not bounded and there exists an $ϵ>0$ such that ${\int }_{{\mathrm{\Omega }}^{-}}{a}^{-}(x,t)<ϵ$. Then there exists an $R_n>0$ such that

$$I(u)\le 0\mathit{\text{for }}u\in H_n\mathrm{\setminus }{B}_{R_n}.$$

(4.1)

Proof Let us choose $\psi \in H$ such that $\parallel \psi \parallel =1$, $\psi \ge 0$ in Ω and $supp(\psi )\subset {\mathrm{\Omega }}^{+}$. Then, by (g2), (g4), and the Hölder inequality, we have

$$\begin{array}{rcl}I(t\psi )& =& \frac{1}{2}{\parallel P_{+}t\psi \parallel }^2-\frac{1}{2}{\parallel P_{-}t\psi \parallel }^2-{\int }_{\mathrm{\Omega }}a(x)G(t\psi )dx\\ =& \frac{1}{2}{\parallel P_{+}t\psi \parallel }^2-\frac{1}{2}{\parallel P_{-}t\psi \parallel }^2-{\int }_{{\mathrm{\Omega }}^{+}}{a}^{+}(x)G(t\psi )dx+{\int }_{{\mathrm{\Omega }}^{-}}{a}^{-}(x)G(t\psi )dx\\ \le & \frac{1}{2}{\parallel P_{+}t\psi \parallel }^2-\frac{1}{2}{\parallel P_{-}t\psi \parallel }^2-{\int }_{{\mathrm{\Omega }}^{+}}{a}^{+}(x)(a_5{t}^{\mu }{\psi }^{\mu }-a_4)dx\\ +{\parallel G(t\psi )\parallel }_{\mathrm{\infty }}{\int }_{{\mathrm{\Omega }}^{-}}{a}^{-}(x)dx\\ \le & \frac{1}{2}{t}^2-{\int }_{{\mathrm{\Omega }}^{+}}{a}^{+}(x)(a_5{t}^{\mu }{\psi }^{\mu }-a_4)dx+{ϵ}^{\prime }\end{array}$$

for small ${ϵ}^{\prime }>0$. Since $\mu >2$, there exist $t_n$ great enough for each n and an $R_n>0$ such that $u_n=t_n\psi$ and $I(u_n)<0$ if $u_n\in H_n\mathrm{\setminus }{B}_{R_n}$ and $\parallel u_n\parallel >R_n$, so the lemma is proved. □

Let us set

$${\mathrm{\Gamma }}_n=\{\gamma \in C([0,1],H)\mid \gamma (0)=0\text{ and }\gamma (1)=u_n\}$$

and

$$b_n=\underset{\gamma \in {\mathrm{\Gamma }}_n}{inf}\underset{[0,1]}{max}I(\gamma (u)),n\in N.$$

Proof of Theorem 1.2(ii) We assume that $g(u)u-\mu G(u)$ is not bounded and there exists an $ϵ>0$ such that ${\int }_{{\mathrm{\Omega }}^{-}}{a}^{-}(x)dx<ϵ$. By Proposition 2.1 and Proposition 2.2, $I\in C^1(H,R)$ and satisfies the Palais-Smale condition. By Lemma 4.1, there exists an $R_n>0$ such that $I(u_n)\le 0$ for $u_n\in H_n\mathrm{\setminus }{B}_{R_n}$. We note that $I(0)=0$. By Lemma 4.1 and the generalized mountain pass theorem, for n large enough $b_n>0$ is a critical value of I and ${lim}_{n\to \mathrm{\infty }}{b}_n=+\mathrm{\infty }$. Let ${\tilde{u}}_n$ be a critical point of I such that $I({\tilde{u}}_n)=b_n$. Then for each real number M, ${max}_{\mathrm{\Omega }}|{\tilde{u}}_n(x)|\ge M$. In fact, by contradiction, ${\mathrm{\Delta }}^{2}u+c\mathrm{\Delta }u=a(x)g(u)$ and ${max}_{\mathrm{\Omega }}|{\tilde{u}}_n(x)|\le K$ imply that

$$I({\tilde{u}}_n)\le \underset{|{\tilde{u}}_n|\le K}{max}(\frac{1}{2}g({\tilde{u}}_n){\tilde{u}}_n-G({\tilde{u}}_n)){\int }_{\mathrm{\Omega }}|a(x)|dx,$$

which means that $b_n$ is bounded. This is absurd because of the fact that ${lim}_{n\to \mathrm{\infty }}{b}_n=+\mathrm{\infty }$. Thus we complete the proof. □