# Existence of solutions for a class of biharmonic equations with the Navier boundary value condition

- Ruichang Pei
^{1}Email author

**2012**:154

https://doi.org/10.1186/1687-2770-2012-154

© Pei; licensee Springer 2012

**Received: **18 July 2012

**Accepted: **14 December 2012

**Published: **28 December 2012

## Abstract

In this paper, the existence of at least one nontrivial solution for a class of fourth-order elliptic equations with the Navier boundary value conditions is established by using the linking methods.

## Keywords

## 1 Introduction

where ${\mathrm{\u25b3}}^{2}$ is the biharmonic operator, $l\in R$ and Ω is a bounded smooth domain in ($N>4$).

The conditions imposed on $f(x,t)$ are as follows:

_{1}) , and there are constants ${C}_{1},{C}_{2}\ge 0$ such that

where ${p}^{\ast}=\frac{2N}{N-4}$;

(H_{2}) $f(x,t)=\circ (|t|)$, $|t|\to 0$, uniformly on Ω;

(H_{3}) ${lim}_{|t|\to \mathrm{\infty}}\frac{f(x,t)}{t}=+\mathrm{\infty}$ uniformly on Ω;

_{4}) There is a constant $\theta \ge 1$ such that for all $(x,t)\in \mathrm{\Omega}\times R$ and $s\in [0,1]$,

where $F(x,t)={\int}_{0}^{t}f(x,s)\phantom{\rule{0.2em}{0ex}}ds$;

_{5}) For some $\delta >0$, either

This fourth-order semilinear elliptic problem has been studied by many authors. In [1], there was a survey of results obtained in this direction. In [2], Micheletti and Pistoia showed that (1.1) admits at least two solutions by a variation of linking if $f(x,u)$ is sublinear. And in [3], the authors proved that the problem (1.1) has at least three solutions by a variational reduction method and a degree argument. In [4], Zhang and Li showed that (1.1) admits at least two nontrivial solutions by the Morse theory and local linking if $f(x,u)$ is superlinear and subcritical on *u*. In [5], Zhang and Wei obtained the existence of infinitely many solutions for the problem (1.1) where the nonlinearity involves a combination of superlinear and asymptotically linear terms. As far as the problem (1.1) is concerned, existence results of sign-changing solutions were also obtained. See, *e.g.*, [6, 7] and the references therein.

We will use linking methods to give the existence of at least one nontrivial solution for (1.1).

*X*be a Banach space with a direct sum decomposition

It is clear that 0 is a critical point of *I*.

The notion of local linking generalizes the notions of local minimum and local maximum. When 0 is a non-degenerate critical point of a functional of class ${C}^{2}$ defined on a Hilbert space and $I(0)=0$, *I* has local linking at 0.

The cases (a), (b) and (c) were considered in [10] with only conditions (1.2) and (1.3), and three theorems about critical points were proved. Using these theorems, the author in [10] proved the existence of at least one nontrivial solution for the second-order elliptic boundary value problem with the Dirichlet boundary value condition. In the present paper, we also use the three theorems in [10] and the linking technique to give the existence of at least one nontrivial solution for the biharmonic problem (1.1) under a weaker condition. The main results are as follows.

**Theorem 1.1** *Assume the conditions* (H_{1})-(H_{4}) *hold*. *If* *l* *is an eigenvalue of* −△ (*with the Dirichlet boundary condition*), *assume also* (H_{5}). *Then the problem* (1.1) *has at least one nontrivial solution*.

We also consider asymptotically quadratic functions.

Let $0<{\lambda}_{1}<{\lambda}_{2}<\cdots <{\lambda}_{k}<\cdots $ be the eigenvalues of $(-\mathrm{\u25b3},{H}_{0}^{1}(\mathrm{\Omega}))$. Then ${\mu}_{j}$ ($j\in {N}_{+}$) is the eigenvalue of $({\mathrm{\u25b3}}^{2}+l\mathrm{\u25b3},{H}^{2}(\mathrm{\Omega})\cap {H}_{0}^{1}(\mathrm{\Omega}))$, where ${\mu}_{j}={\lambda}_{j}({\lambda}_{j}-l)$. We assume that

(H_{6}) $f(x,u)={f}_{\mathrm{\infty}}u+\circ (|u|)$, $|u|\to \mathrm{\infty}$, uniformly in Ω, and ${\mu}_{k}<{f}_{\mathrm{\infty}}<{\mu}_{k+1}$.

**Theorem 1.2** *Assume the conditions* (H_{1}), (H_{6}) *and one of the following conditions*:

(A_{1}) ${\lambda}_{j}<l<{\lambda}_{j+1}$, $j\ne k$;

_{2}) ${\lambda}_{j}=l<{\lambda}_{j+1}$, $j\ne k$,

*for some*$\delta >0$,

_{3}) ${\lambda}_{j}<l={\lambda}_{j+1}$, $j\ne k$,

*for some*$\delta >0$,

*Then the problem* (1.1) *has at least one nontrivial solution*.

## 2 Preliminaries

*X*be a Banach space with a direct sum decomposition

A sequence $({\alpha}_{n})\subset {N}^{2}$ is admissible if for every $\alpha \in {N}^{2}$, there is $m\in N$ such that $n\ge m\Rightarrow {\alpha}_{n}\ge \alpha $. For every $I:X\to R$, we denote by ${I}_{\alpha}$ the function *I* restricted ${X}_{\alpha}$.

**Definition 2.1**Let

*I*be locally Lipschitz on

*X*and $c\in R$. The functional

*I*satisfies the ${(C)}_{c}^{\ast}$ condition if every sequence $({u}_{{\alpha}_{n}})$ such that $({\alpha}_{n})$ is admissible and

contains a subsequence which converges to a critical point of *I*.

**Definition 2.2**Let

*I*be locally Lipschitz on

*X*and $c\in R$. The functional

*I*satisfies the ${(C)}^{\ast}$ condition if every sequence $({u}_{{\alpha}_{n}})$ such that $({\alpha}_{n})$ is admissible and

contains a subsequence which converges to a critical point of *I*.

**Remark 2.1**

- 1.
The ${(C)}^{\ast}$ condition implies the ${(C)}_{c}^{\ast}$ condition for every $c\in R$.

- 2.
When the ${(C)}_{c}^{\ast}$ sequence is bounded, then the sequence is a ${(PS)}_{c}^{\ast}$ sequence (see [11]).

- 3.Without loss of generality, we assume that the norm in
*X*satisfies${\parallel {u}_{1}+{u}_{2}\parallel}^{2}={\parallel {u}_{1}\parallel}^{2}+{\parallel {u}_{2}\parallel}^{2},\phantom{\rule{1em}{0ex}}{u}_{j}\in {X}_{j},j=1,2.$

**Definition 2.3**Let

*X*be a Banach space with a direct sum decomposition

**Lemma 2.1** (see [10])

*Suppose that* $I\in {C}^{1}(X,R)$ *satisfies the following assumptions*:

(B_{1}) *I* *has a local linking at* 0 *and* ${X}^{1}\ne \{0\}$;

(B_{2}) *I* *satisfies* ${(PS)}^{\ast}$;

(B_{3}) *I* *maps bounded sets into bounded sets*;

(B_{4}) *for every* $m\in N$, $I(u)\to -\mathrm{\infty}$, $\parallel u\parallel \to \mathrm{\infty}$, $u\in X={X}_{m}^{1}\oplus {X}^{2}$. *Then* *I* *has at least two critical points*.

**Remark 2.2** Assume *I* satisfies the ${(C)}_{c}^{\ast}$ condition. Then this theorem still holds.

*X*be a real Hilbert space and let $I\in {C}^{1}(X,R)$. The gradient of

*I*has the form

where *A* is a bounded self-adjoint operator, 0 is not the essential spectrum of *A*, and *B* is a nonlinear compact mapping.

by ${p}_{\alpha}:X\to {X}_{\alpha}$ the orthogonal projector onto ${X}_{\alpha}$, and by ${M}^{-}(L)$ the Morse index of a self-adjoint operator *L*.

**Lemma 2.2** (see [10])

*I*

*satisfies the following assumptions*:

- (i)
*I**has a local linking at*0*with respect to*$({X}^{1},{X}^{2})$; - (ii)
*there exists a compact self*-*adjoint operator*${B}_{\mathrm{\infty}}$*such that*$B(u)={B}_{\mathrm{\infty}}(u)+\circ (\parallel u\parallel ),\phantom{\rule{1em}{0ex}}\parallel u\parallel \to \mathrm{\infty};$ - (iii)
$A+{B}_{\mathrm{\infty}}$

*is invertible*; - (iv)
*for infinitely many multiple*-*indices*$\alpha :=(n,n)$,${M}^{-}((A+{P}_{\alpha}{B}_{\mathrm{\infty}}){|}_{{X}_{\alpha}})\ne dim{X}_{n}^{2}.$

*Then* *I* *has at least two critical points*.

## 3 The proof of main results

*Proof of Theorem 1.1*(1) We shall apply Lemma 2.1 to the functional

Then other case is similar and simple.

*X*. Choose a Hilbertian basis ${e}_{n}$ ($n\ge 0$) for

*X*and define

By the condition (H_{1}) and Sobolev inequalities, it is easy to see that the functional *I* belongs to ${C}^{1}(X,R)$ and maps bounded sets to bounded sets.

*I*has a local linking at 0 with respect to $({X}^{1},{X}^{2})$. Decompose ${X}^{1}$ into $V+W$ when $V=ker(-{\mathrm{\u25b3}}^{2}+l\mathrm{\u25b3})$, $W={({X}^{2}+V)}^{\mathrm{\perp}}$. Also, set $u=v+w$, $u\in {X}^{1}$, $v\in V$, $w\in W$. By the equivalence of norm in the finite-dimensional space, there exists $C>0$ such that

_{1}) and (H

_{2}) that for any $\u03f5>0$, there exists ${C}_{\u03f5}$ such that

_{5}), we obtain

for $0<r$ small enough.

*I*satisfies ${(C)}_{c}^{\ast}$. Consider a sequence $({u}_{{\alpha}_{n}})$ such that $({u}_{{\alpha}_{n}})$ is admissible and

*n*large enough, $2\sqrt{m}{\parallel {u}_{{\alpha}_{n}}\parallel}^{-1}\in (0,1)$, and combining Ehrling-Nirenberg-Gagliardo inequality, we have

where *ϵ* is a small enough constant.

This contradicts (3.5).

_{3}), we have

By (3.8), the right-hand side of (3.9) $\to +\mathrm{\infty}$. This is a contradiction.

In any case, we obtain a contradiction. Therefore, $\{{u}_{{\alpha}_{n}}\}$ is bounded.

_{2}) and (H

_{3}), there exists large enough

*M*such that

Hence, our claim holds. □

*Proof of Theorem 1.2* We omit the proof which depends on Lemma 2.2 and is similar to the preceding one. □

## Author’s contributions

The author read and approved the final manuscript.

## Declarations

### Acknowledgements

The author would like to thank the referees for valuable comments and suggestions in improving this paper. This work was supported by the National NSF (Grant No. 10671156) of China.

## Authors’ Affiliations

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