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# Fourth order elliptic boundary value problem with nonlinear term decaying at the origin

- Tacksun Jung
^{1}and - Q-Heung Choi
^{2}Email author

**2013**:432

https://doi.org/10.1186/1029-242X-2013-432

© Jung and Choi; licensee Springer. 2013

**Received:**24 April 2013**Accepted:**31 July 2013**Published:**13 September 2013

## Abstract

We consider the number of the weak solutions for some fourth order elliptic boundary value problem with bounded nonlinear term decaying at the origin. We get a theorem, which shows the existence of the bounded solution for this problem. We obtain this result by approaching the variational method and using the generalized mountain pass theorem for the fourth order elliptic problem with bounded nonlinear term.

**MSC:**35J30, 35J40.

## Keywords

- fourth order elliptic boundary value problem
- nonlinear term decaying at the origin
- bounded nonlinear term
- variational method
- generalized mountain pass theorem
- ${(\mathit{PS})}_{c}$ condition

## 1 Introduction

*∂*Ω. Let $c\in {R}^{1}$ and $g:\overline{\mathrm{\Omega}}\times R\to R$ be a ${C}^{1}$ function. In this paper, we consider the number of the weak solutions for the following fourth order elliptic problem with the Dirichlet boundary condition

We assume that $g\in {C}^{1}(\overline{\mathrm{\Omega}}\times R,R)$ satisfies the following:

(g1) $g\in {C}^{1}(\overline{\mathrm{\Omega}}\times R,R)$,

(g2) $g(x,0)=0$, $g(x,\xi )=o(|\xi |)$ uniformly with respect to $x\in \overline{\mathrm{\Omega}}$,

(g3) there exists $C>0$ such that $|g(x,\xi )|<C$$\mathrm{\forall}(x,\xi )\in \overline{\mathrm{\Omega}}\times R$.

Furthermore, we assume that $c\in {R}^{1}$ satisfies ${\lambda}_{j}<c<{\lambda}_{j+1}$.

Jung and Choi [1] proved that (1.1) has at least one nontrivial solution, when $c<{\lambda}_{1}$ and *g* satisfies the condition (g1), (g2) and additional conditions

(g3)′ there exists $\xi \ge 0$ such that $p(x,\xi )\le 0$$\mathrm{\forall}x\in \overline{\mathrm{\Omega}}$,

(g4)′ there exist a constant $r>0$ and an element $e\in H$ such that $\parallel e\parallel =r$, $e<\xi $ and $\frac{1}{2}{r}^{2}-{\int}_{\mathrm{\Omega}}P(x,e)<0$,

She showed that if $c<{\lambda}_{1}$ and $b\ge {\mathrm{\Lambda}}_{1}$, then (1.3) has a negative solution. She obtained this result by the degree theory. Micheletti and Pistoia [6] also proved that if $c<{\lambda}_{1}$ and $b\ge {\mathrm{\Lambda}}_{2}$, then (1.3) has at least three solutions by the variational linking theorem and Leray-Schauder degree theory.

where *H* is introduced in Section 2.

where $G(x,s)={\int}_{0}^{s}g(x,\tau )\phantom{\rule{0.2em}{0ex}}d\tau $. By (g1), *I* is well defined.

Our main result is the following.

**Theorem 1.1***Assume that*${\lambda}_{j}<c<{\lambda}_{j+1}$, $j\ge 1$, *and**g**satisfies the conditions* (g1)-(g3). *Then* (1.1) *has at least one bounded weak solution*.

We prove Theorem 1.1 by approaching the variational method and using the mountain pass theorem for the reduced fourth order elliptic problem with bounded nonlinear term. The outline of the proof of Theorem 1.1 is as follows: In Section 2, we prove that functional $I(u)\in {C}^{1}$ and the functional *I* satisfies the Palais-Smale condition. In Section 3, we show that the functional *I* satisfies the generalized mountain pass theorem, and so, prove that *I* has at least one nontrival critical point, from which we prove Theorem 1.1.

## 2 Variational approach

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

*H*of ${L}^{2}(\mathrm{\Omega})$ as follows

*c*is fixed, we have ${\mathrm{\Lambda}}_{k}\to \mathrm{\infty}$ and

- (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 is proved in [7].

Then $H={H}_{-}\oplus {H}_{+}$, for $u\in H$, $u={u}^{-}+{u}^{+}\in {H}_{-}\oplus {H}_{+}$. Let ${P}_{+}$ be the orthogonal projection from *H* onto ${H}_{+}$ and ${P}_{-}$ be the orthogonal projection from *H* onto ${H}_{-}$. We can write ${P}_{+}u={u}^{+}$, ${P}_{-}u={u}^{-}$, for $u\in H$.

By the following Lemma 2.1, the weak solutions of (1.1) coincide with the critical points of the associated functional $I(u)$.

**Lemma 2.1**

*Assume that*${\lambda}_{j}<c<{\lambda}_{j+1}$, $j\ge 1$,

*and*

*g*

*satisfies the conditions*(g1)-(g3).

*Then*$I(u)$

*is continuous*,

*and Fréchet differentiable in*

*H*

*with Fréchet derivative*

*If we set*

*then*${F}^{\prime}(u)$

*is continuous with respect to weak convergence*, ${F}^{\prime}(u)$

*is compact and*

*this implies that*$I\in {C}^{1}(H,R)$*and*$F(u)$*is weakly continuous*.

The proof of Lemma 2.1 has the similar process to that of the proof in Appendix B in [8].

Now, we shall show that $I(u)$ satisfies the Palais-Smale condition.

**Lemma 2.2***Assume that*${\lambda}_{j}<c<{\lambda}_{j+1}$, $j\ge 1$, *and**g**satisfies the conditions* (g1)-(g3). *Then the functional**I**satisfies the Palais*-*Smale condition*: *Any sequence*$({u}_{m})$*in**H*, *for which*$|I({u}_{m})|\le M$*and*${I}^{\prime}({u}_{m})\to 0$*as*$m\to \mathrm{\infty}$, *possesses a convergent subsequence*.

*Proof*Let us choose $u\in H$. By $g\in {C}^{1}$ and (g1), $G(x,u)$ is bounded. Then we have

Since *u* is bounded and ${\int}_{\mathrm{\Omega}}G(x,u)\phantom{\rule{0.2em}{0ex}}dx$ is bounded, $I(u)$ is bounded from below. Thus, *I* satisfies the $(\mathit{PS})$ condition. □

## 3 Proof of Theorem 1.1

Now, we recall the generalized mountain pass theorem (*cf.* Theorem 5.3 in [8]).

**Theorem 3.1** (Generalized mountain pass theorem)

*Let*

*H*

*be a real Banach space with*$H=V\oplus X$,

*where*$V\ne \{0\}$

*and is finite*-

*dimensional*.

*Suppose that*$I\in {C}^{1}(H,R)$

*satisfies*$(\mathit{PS})$

*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*

*where*

*We shall show that the functional**I**satisfies the generalized mountain pass geometrical assumptions*.

*Let*${H}_{j}=span\{{\varphi}_{1},\dots ,{\varphi}_{j}\}$.

*Then*${H}_{j}$

*is a subspace of*

*H*

*such that*

*Let*

**Lemma 3.1**

*Assume that*${\lambda}_{j}<c<{\lambda}_{j+1}$

*and*

*g*

*satisfies*(g1)-(g3).

*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}_{j}^{\perp}}\ge \alpha $,*and* - (ii)
*there is an*$e\in \partial {B}_{1}\cap {H}_{j}^{\perp}$*and*$R>\rho $*such that if*$Q=({\overline{B}}_{R}\cap {H}_{j})\oplus \{re\mid 0<r<R\}$,*then*$I{|}_{\partial Q}\le 0$,*and* - (iii)
*there exists*${u}_{0}\in H\mathrm{\setminus}Q$*such that*$\parallel {u}_{0}\parallel >R$*and*$I({u}_{0})\le 0$.

*Proof*(i) Let $u\in {H}_{j}^{\perp}$. We note that

- (ii)Let us choose an element $e\in \partial {B}_{1}\cap {H}_{j}^{\perp}$. Let $u\in (\overline{{B}_{r}}\cap {H}_{j})\oplus \{re\mid 0<r\}$. Then $u=v+w$, $v\in {B}_{r}\cap {H}_{j}$, $w=re$. We note that$\text{if}v\in {B}_{r}\cap {H}_{j},\phantom{\rule{1em}{0ex}}{\int}_{\mathrm{\Omega}}({\mathrm{\Delta}}^{2}v+c\mathrm{\Delta}v)v\phantom{\rule{0.2em}{0ex}}dx\le {\lambda}_{j}({\lambda}_{j}-c){\parallel v\parallel}_{{L}^{2}(\mathrm{\Omega})}^{2}0.$

- (iii)
If we choose ${u}_{0}\in H\mathrm{\setminus}Q$, then by (ii), $I({u}_{0})\le 0$. □

*Proof of Theorem 1.1*We will show that $I(u)$ has a nontrivial critical point by the generalized mountain pass theorem. By Lemma 2.1, $I(u)$ is continuous and Fréchet differentiable in

*H*. By Lemma 2.2, the functional

*I*satisfies $(\mathit{PS})$ condition. We note that $I(0)=0$. 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}_{j}^{\perp}}\ge \alpha $, and there is an $e\in \partial {B}_{1}\cap {H}_{j}^{\perp}$ and $R>\rho $ such that if $Q=({\overline{B}}_{R}\cap {H}_{j})\oplus \{re\mid 0<r<R\}$. Let us set

*I*possesses a critical value $b\ge \alpha $. Moreover,

*b*can be characterized as

*I*has at least one nontrivial critical point. We denote by $\tilde{u}$ a critical point of

*I*such that $I(\tilde{u})=b$. We claim that

*b*is bounded. In fact, by (iii) of Lemma 3.1, we have

*b*is bounded:

is not bounded, which is absurd to the fact that $b=I(\tilde{u})$ is bounded. Thus, $\tilde{u}$ is bounded, so (1.1) has at least one bounded weak solution. Thus, we prove Theorem 1.1. □

## Declarations

### Acknowledgements

This work (Tacksun Jung) was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (KRF-2010-0023985).

## Authors’ Affiliations

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