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# Multiple solutions for a fourth-order nonlinear elliptic problem which is superlinear at +∞ and linear at −∞

- Ruichang Pei
^{1, 2}Email author and - Jihui Zhang
^{2}

**2014**:12

https://doi.org/10.1186/1687-2770-2014-12

© Pei and Zhang; licensee Springer. 2014

**Received:**14 August 2013**Accepted:**16 December 2013**Published:**10 January 2014

## Abstract

We consider a semilinear fourth-order elliptic equation with a right-hand side nonlinearity which exhibits an asymmetric growth at +∞ and at −∞. Namely, it is linear at −∞ and superlinear at +∞. Combining variational methods with Morse theory, we show that the problem has at least two nontrivial solutions, one of which is negative.

## Keywords

- fourth-order elliptic boundary value problems
- multiple solutions
- critical groups
- Morse theory

## 1 Introduction

where ${\mathrm{\u25b3}}^{2}$ is the biharmonic operator, and Ω is a bounded smooth domain in ${\mathbb{R}}^{N}$ ($N>4$), and $c<{\lambda}_{1}^{\ast}$ the first eigenvalue of −△ in ${H}_{0}^{1}(\mathrm{\Omega})$.

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

(H_{1}) $f\in {C}^{1}(\overline{\mathrm{\Omega}}\times \mathbb{R},\mathbb{R})$, $f(x,0)=0$ for all $x\in \mathrm{\Omega}$ and $f(x,t)<0$ for all $t<0$ and all $x\in \mathrm{\Omega}$;

(H_{2}) there exist $r\in (2,{p}^{\ast})$ and $A,B>0$ such that $|{f}_{t}^{\prime}(x,t)|\le A+B{|t|}^{r-2}$ for all $x\in \mathrm{\Omega}$, and $t\in \mathbb{R}$, where ${p}^{\ast}=\frac{2N}{N-4}$, if $N>4$;

(H_{3}) ${lim}_{t\to -\mathrm{\infty}}\frac{f(x,t)}{t}=l$ uniformly for $x\in \mathrm{\Omega}$, where *l* is a nonnegative constant;

_{4}) there exist $\beta ,\xi \in \mathbb{R}$ such that for $F(x,t)={\int}_{0}^{t}f(x,s)\phantom{\rule{0.2em}{0ex}}ds$, we have

_{5}) there exist ${\vartheta}_{1},{\vartheta}_{2}\in {L}^{\mathrm{\infty}}{(\mathrm{\Omega})}_{+}$ and an integer $k\ge 2$ such that

_{3}) and equation (3) in (H

_{4}), it is clear that for all $x\in \mathrm{\Omega}$, $f(x,t)$ is linear at −∞ and superlinear at +∞. Clearly, $u=0$ is a trivial solution of problem (1). It follows from (H

_{1}) and (H

_{2}) that the functional

_{2}), the critical points of

*I*are solutions of problem (1). Let $0<{\lambda}_{1}<{\lambda}_{2}<\cdots <{\lambda}_{k}<\cdots $ be the eigenvalues of $({\mathrm{\u25b3}}^{2}+c\mathrm{\u25b3},{H}^{2}(\mathrm{\Omega})\cap {H}_{0}^{1}(\mathrm{\Omega}))$ and ${\varphi}_{1}(x)>0$ be the eigenfunction corresponding to ${\lambda}_{1}$. In fact, ${\lambda}_{1}={\lambda}_{1}^{\ast}({\lambda}_{1}^{\ast}-c)$. Let ${E}_{{\lambda}_{k}}$ denote the eigenspace associated with ${\lambda}_{k}$. Throughout this article, we denoted by ${|\cdot |}_{p}$ the ${L}^{p}(\mathrm{\Omega})$ norm and $E={H}^{2}(\mathrm{\Omega})\cap {H}_{0}^{1}(\mathrm{\Omega})$. The aim of this paper is to prove a multiplicity theorem for problem (1) when the nonlinearity term $f(x,t)$ exhibits an asymmetric behavior as $t\in \mathbb{R}$ approaches +∞ and −∞. In the past, some authors studied the following elliptic problem:

with asymmetric nonlinearities by using the Fučík spectrum of the operator $(-\mathrm{\u25b3},{H}_{0}^{1}(\mathrm{\Omega}))$. This approach requires that $f(x,t)$ exhibits linear growth at both +∞ and −∞ and that the limits ${lim}_{t\to \pm \mathrm{\infty}}\frac{f(x,t)}{t}$ exist and belong to ℝ. See the works of Các [1], Dancer and Zhang [2], Magalhães [3], de Paiva [4], Schechter [5] and the references therein. Equations with nonlinearities which are superlinear in one direction and linear in the other were investigated by Arcoya and Villegas [6] and Perera [7]. They let the nonlinearity $f(x,t)$ be line at −∞ and satisfy the Ambrosetti-Rabinowitz condition at +∞. Particularly, it is worth noticing paper [8]. The authors relax several of the above restrictions on the nonlinearity $f(x,t)$. Their nonlinearity is only measurable in $x\in \mathrm{\Omega}$. The limit as $t\to -\mathrm{\infty}$ of $\frac{f(x,t)}{t}$ need not exist and the growth at −∞ can be linear or sublinear. Furthermore, their nonlinearity $f(x,t)$ does not satisfy the famous AR-condition. They use the truncated skill of first order weak derivative to verify the (PS) condition and obtain multiple solutions for problem (1) by combining variational methods and Morse theory.

To the authors’ knowledge, there seem to be few results on problem (1) when $f(x,t)$ is asymmetric nonlinearity at positive infinity and at negative infinity. However, the method in [8] cannot be applied directly to the biharmonic problems. For example, for the Laplacian problem, $u\in {H}_{0}^{1}(\mathrm{\Omega})$ implies $|u|,{u}_{+},{u}_{-}\in {H}_{0}^{1}(\mathrm{\Omega})$, where ${u}_{+}=max(u,0)$, ${u}_{-}=max(-u,0)$. We can use ${u}_{+}$ or ${u}_{-}$ as a test function, which is helpful in proving a solution nonnegative. While for the biharmonic problems, this trick fails completely since $u\in {H}_{0}^{2}(\mathrm{\Omega})$ does not imply ${u}_{+},{u}_{-}\in {H}_{0}^{2}(\mathrm{\Omega})$ (see [[9], Remark 2.1.10] and [10, 11]). As far as this point is concerned, we will make use of the new methods to overcome it.

This fourth-order semilinear elliptic problem can be considered as an analogue of a class of second-order problems which have been studied by many authors. In [12], there was a survey of results obtained in this direction. In [13], Micheletti and Pistoia showed that $({P}_{1})$ admits at least two solutions by a variation of linking if $f(x,u)$ is sublinear. Chipot [14] proved that the problem $({P}_{1})$ has at least three solutions by a variational reduction method and a degree argument. In [15], Zhang and Li showed that $({P}_{1})$ admits at least two nontrivial solutions by Morse theory and local linking if $f(x,u)$ is superlinear and subcritical on *u*.

In this article, under the guidance of [8], we consider multiple solutions of problem (1) with the asymmetric nonlinearity by using variational methods and Morse theory.

## 2 Main result and auxiliary lemmas

Let us now state the main result.

**Theorem 2.1** *Assume conditions* (H_{1})-(H_{5}) *hold*. *If* $l<{\lambda}_{1}$, *then problem* (1) *has at least two nontrivial solutions*.

**Lemma 2.2** *Under the assumptions of Theorem * 2.1, *then* *I* *satisfies the* (*PS*) *condition*.

*Proof*Let $\{{u}_{n}\}\subset E$ be a sequence such that for every $n\in N$,

for all $v\in E$.

_{3}) and equation (3) in (H

_{4}), we have

*f*, we also deduce

which contradicts inequality (16). Thus $|{\mathrm{\Omega}}^{+}|=0$ and the claim is proved.

_{3}), there exists $C>0$ such that $\frac{|f(x,{u}_{n})|}{|{u}_{n}|}\le C$ for a.e. $x\in \mathrm{\Omega}$. By using Lebesgue dominated convergence theorem in equation (15), we have

for all $v\in E$. This contradicts $l<{\lambda}_{1}$. □

**Lemma 2.3**

*Let*$E=V\oplus W$,

*where*$V={E}_{{\lambda}_{1}}\oplus {E}_{{\lambda}_{2}}\oplus \cdots \oplus {E}_{{\lambda}_{k}}$.

*If*$k\ge 0$

*is an integer*, $\vartheta \in {L}^{\mathrm{\infty}}{(\mathrm{\Omega})}_{+}$, $\vartheta (x)\le {\lambda}_{k+1}$

*a*.

*e*.

*on*Ω

*and the inequality is strict on a set of positive measure*,

*then there exists*$\gamma >0$

*such that*

*for all* $u\in W$.

*Proof*We claim that there exists a constant ${\vartheta}_{0}<1$ such that

*n*. By the homogeneity of the above inequality, we may assume that $\parallel {u}_{n}\parallel =1$ and

*n*. It follows from the weak compactness of the unit ball of

*W*that there exists a subsequence, say $\{{u}_{n}\}$, such that ${u}_{n}$ weakly converges to

*u*in

*W*. Now Sobolev’s embedding theorem suggests that $\{{u}_{n}\}$ converges to

*u*in ${L}^{2}(\mathrm{\Omega})$. From inequality (19) we obtain

which implies that $u\in {E}_{{\lambda}_{k+1}}\setminus \{0\}$ and $u=0$ on a positive measure subset. It contradicts the unique continuation property of the eigenfunction. □

## 3 Computation of the critical groups

It is well known that critical groups and Morse theory are the main tools in solving elliptic partial differential equation. Let us recall some results which will be used later. We refer the readers to the book [16] for more information on Morse theory.

*H*be a Hilbert space and $I\in {C}^{1}(H,\mathbb{R})$ be a functional satisfying the (PS) condition or (C) condition, and ${H}_{q}(X,Y)$ be the

*q*th singular relative homology group with integer coefficients. Let ${u}_{0}$ be an isolated critical point of

*I*with $I({u}_{0})=c$, $c\in \mathbb{R}$, and

*U*be a neighborhood of ${u}_{0}$. The group

is said to be the *q* th critical group of *I* at ${u}_{0}$, where ${I}^{c}=\{u\in H:I(u)\le c\}$.

*I*and $a<infI(K)$, the critical groups of

*I*at infinity are formally defined by [17]

For the convenience of our proof, we first recall two interesting results and prove two important propositions.

**Proposition 3.1** [18]

*Under* (H_{2}), *if* $u\in E:={H}^{2}(\mathrm{\Omega})\cap {H}_{0}^{1}(\mathrm{\Omega})$ *is an isolated critical point of* *I*, *then* ${C}_{\ast}(I,u)\cong {C}_{\ast}(I{|}_{{C}_{0}^{3}(\mathrm{\Omega})},u)$.

**Proposition 3.2** [19]

*If* ${D}_{1}\subset D\subset {E}_{0}\subset {E}_{1}\subset X$ *and for some integer* $k\ge 0$ *we have* ${H}_{k}(E,D)\ne 0$ *and* ${H}_{k}({E}_{1},{D}_{1})=0$, *then either* ${H}_{k+1}({E}_{1},E)\ne 0$ *or* ${H}_{k-1}(D,{D}_{1})\ne 0$.

**Proposition 3.3**

*If the assumptions of Theorem*2.1

*hold*,

*then*

*Proof*Under the guidance of [8] and [18], we begin to prove this result. Let ${I}_{1}=I{|}_{{C}_{0}^{3}(\overline{\mathrm{\Omega}})}$. Indeed, it follows from above Proposition 3.1 that

*I*and ${I}_{1}$ have same critical set. Since ${C}_{0}^{3}(\overline{\mathrm{\Omega}})$ is dense in

*E*, invoking Proposition 16 of Palais [20], we have

Clearly, ${h}_{+}$ is a continuous homotopy and $h(1,u)={\varphi}_{1}$ for all $x\in \partial {B}_{1,+}^{c}$. Therefore, $\partial {B}_{1,+}^{c}$ is contractible in itself.

_{4}), given any $\gamma >0$, we can find $C=C(\gamma )>0$ such that

_{3}), and by choosing $C>0$ even bigger if necessary, we observe that there is a number ${\gamma}_{0}>0$ such that

_{2}), we have

for some ${C}_{3}>0$.

_{4}), we see that there exist constants ${\xi}_{0}$ and $M>0$ such that

_{2}) and formula (2) in condition (H

_{4}), we have

*C*is a positive constant. Let $i:{C}_{0}^{3}(\overline{\mathrm{\Omega}})\to E$ be the continuous embedding map. Let ${\u3008\cdot ,\cdot \u3009}_{0}$ denote the duality brackets for the pair $({C}_{0}^{3}{(\overline{\mathrm{\Omega}})}^{\ast},{C}_{0}^{3}(\overline{\mathrm{\Omega}}))$. We let ${I}_{2}=I\circ i$, and so

_{2}) and (H

_{3}), we see that given $\u03f5>0$, we can find $M>0$ such that

Moreover, the implicit function theorem implies that $k\in C(\partial {B}_{1,+}^{c},[1,+\mathrm{\infty}))$.

*a*, we have

Combining with equation (36) leads to equation (22), which completes the proof. □

**Proposition 3.4**

*If the assumptions of Theorem*2.1

*hold*,

*then*

*where* $d=dimV$ (*V* *being defined in Lemma * 2.3).

*Proof*By condition (H

_{5}), given $\u03f5>0$, we can find ${\delta}_{\ast}>0$ such that

*V*is finite dimensional, all norms are equivalent. Thus we can find $\rho >0$ small such that

for all $u\in V$ and $\parallel u\parallel \le \rho $.

_{2}) and (H

_{5}) that

*ρ*small enough we have

From inequalities (40) and (43), we know that *I* has a local linking at 0. Then invoking Proposition 2.3 of Bartsch and Li [17], we obtain ${C}_{d}(I,0)\ne 0$. □

## 4 Proof of the main result

*Proof of Theorem 2.1*We consider the following problem:

_{1}) and (H

_{3}), we know that ${I}_{-}$ is coercive and boundedness from below. Thus we can find ${v}_{0}\in E$ such that

_{5}), given $\u03f5\in (0,{\lambda}_{k}-{\lambda}_{1})$, there exists $\delta >0$ such that

*s*small enough, it follows from inequality (45) that

_{1}) and strong maximum principle, we have ${v}_{0}<0$ and

*I*. Otherwise, we have a second nontrivial smooth solution and so we are done. By Proposition 3.3, we have

*I*satisfies the (PS) condition (see Lemma 2.2). Hence choosing $\u03f5>0$ small enough, we have

*I*such that

*I*such that

Since $d\ge 2$, from equations (46) and (49), we see that ${v}^{\ast}\ne {v}_{0}$. It is obvious that ${v}^{\ast}\ne 0$. Therefore ${v}_{0}$ and ${v}^{\ast}$ are two solutions of problem (1). □

## Declarations

### Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions in improving this article. This study was supported by the National NSF (Grant No. 11101319) of China and Planned Projects for Postdoctoral Research Funds of Jiangsu Province (Grant No. 1301038C).

## Authors’ Affiliations

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