Research

# Infinitely many sign-changing solutions for p-Laplacian equation involving the critical Sobolev exponent

Yuanze Wu* and Yisheng Huang

Author Affiliations

Department of Mathematics, Soochow University, Suzhou, Jiangsu, 215006, P.R. China

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Boundary Value Problems 2013, 2013:149  doi:10.1186/1687-2770-2013-149

 Received: 14 December 2012 Accepted: 28 May 2013 Published: 19 June 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we study the following problem:

where is a smooth bounded domain, , is the p-Laplacian, is the critical Sobolev exponent and is a parameter. By establishing a new deformation lemma, we show that if , then for each , this problem has infinitely many sign-changing solutions, which extends the results obtained in (Cao et al. in J. Funct. Anal. 262: 2861-2902, 2012; Schechter and Zou in Arch. Ration. Mech. Anal. 197: 337-356, 2010).

### 1 Introduction

In this paper, we consider the following problem:

(1.1)

where () is a smooth bounded domain, , is the p-Laplacian, is the critical Sobolev exponent and is a parameter.

The first existence result of Problem (1.1) with was obtained by Brezis and Nirenberg in the celebrated paper [1]. In that paper, the authors proved that Problem (1.1) had a positive solution for and or and , where is the first eigenvalue of . After that, many existence results have appeared for (1.1); one can see, for example, [2-7] and the references therein for case of and [8-11] and the references therein for case of . In particular, in the case of , the authors in [2] proved that the number of solutions of Problem (1.1) is bounded below by the number of eigenvalues of lying in the open interval , where S is the best Sobolev constant and is the Lebesgue measure of Ω. In [5], the existence of infinitely many sign-changing solutions of (1.1) with has been obtained when , and Ω is a ball, while it has been shown in [6] that (1.1) with has infinitely many sign-changing radial solutions when , and Ω also is a ball. We remark that the methods used in [5,6] are strongly dependent on the symmetry of the ball Ω. For a general bounded smooth domain Ω, by the method of invariant sets of the descending flow, the authors in [7] have shown that (1.1) with has infinitely many sign-changing solutions when and , which extends the main result in [4].

The main purpose of this paper is to try to obtain the existence of infinitely many sign-changing solutions of Problem (1.1) for general . In a very recent work [9], the authors have proved that (1.1) has infinitely many solutions for and . However, by using the Picone identity (cf.[12,13]), we see that every nonzero solution of Problem (1.1) is sign-changing for , where is the first eigenvalue of (see Lemma 2.1 for more details). Hence, to achieve our purpose, we mainly consider the situation of .

Our main result in this paper is the following.

Theorem 1.1Assume thatand. Then Problem (1.1) has infinitely many sign-changing solutions.

Since is the critical Sobolev exponent, in order to overcome the lack of compactness of the embedding of in the Lebesgue space , we follow the ideas of [4,7,9] to consider the following auxiliary problems:

where and is increasing to . It has been shown by [[14], Theorem 1.2] that for every n, Problem () has infinitely many sign-changing solutions . Hence, to prove Theorem 1.1, we will show that for every , converges to some sign-changing solution of (1.1) as , and that are different. The convergence of can be done with the help of [[9], Theorem 1.2], which we show in Lemma 2.3. To distinguish , we shall establish a new deformation lemma on special sets in ; see Lemma 2.5 for details.

Throughout this paper, we will always respectively denote and by the usual norm in spaces and (). Let C be indiscriminately used to denote various positive constants.

### 2 Proof of Theorem 1.1

We first consider the case of . Recall that , the first eigenvalue of in , given by , is simple and there exists a positive eigenfunction corresponding to such that for every (cf.[15]). Moreover, by [[16], Proposition 2.1], we know that . On the other hand, we have the following proposition which is the so-called Picone identity.

Proposition 2.1 [[13], Lemma A.6]

Letbe such that, and. Then

Moreover,

and the equality holds if and only iffor some constant.

Lemma 2.1Assume thatis a nonzero solution of (1.1) for. Thenuis sign-changing.

Proof By a contradiction, we may assume . By using a standard regularity argument and [[17], Lemmas 3.2 and 3.3], we have for some . Thus, it follows from the strong maximum principle (cf.[18]) that . Now, for every , by applying the above Picone identity (i.e., Proposition 2.1) to and , we see

Noting that u is a solution of (1.1), we have

It follows from the Fatou lemma that

which is impossible since , , and . Therefore, we have proved Lemma 2.1. □

Next, we consider the case of .

It is clear that the corresponding functional of () , given by

is Fréchet differentiable. Let , where is a linearly independent sequence of . It is easy to show that there exists such that for , where (cf. [[14], Lemma 3.9]). We denote

Recall that the genus of a symmetric set A of is defined by

Here, we say that A is symmetric if implies .

By [[14], Theorem 1.2], we know that, for every , has infinitely many critical points, denoted by , in for μ small enough. Moreover,

(2.1)

where .

Lemma 2.2For every, there existssuch thatfor all.

Proof Consider the following auxiliary functional:

where . Since , we may assume for all . Then for all . This means

(2.2)

Note that is the corresponding functional of the following equation:

and the nonlinearity satisfies the assumptions of [[14], Theorem 1.2]. Thus, this equation has a sequence of solutions such that

for μ small enough. For every , the definitions of and , together with (2.2), imply for all . On the other hand, since for every n, is a sequence of solutions for () whose energies satisfy (2.1), it follows that . We complete the proof by choosing . □

By Lemma 2.2 and [[9], Theorem 1.2], we know that for each , there exists such that as in . The next lemma will give more information about .

Lemma 2.3is a sign-changing solution of Problem (1.1) for every.

Proof We first prove that is a solution of Problem (1.1) for every . Since as in ,

and

as for every . If we can prove

(2.3)

as for every , then is a solution of (1.1) for is a solution of (). Indeed, a.e. in Ω as since in . By the Egoroff theorem, for every , there exists such that uniformly in and , where is the Lebesgue measure of . This, together with the Lebesgue dominated convergence theorem, implies

(2.4)

On the other hand, for every , we have

For every , by the above inequality and the absolute continuity of the integral, we can take δ small enough such that

For this δ, since in ,

for n large enough. By (2.4), for this δ, we have

for n large enough. So (2.3) holds. Moreover, by a similar proof, we can show .

Next, we will show is sign-changing for all . Since for each , is a sign-changing solution of (), multiplying () by , we obtain , where . Note that , by the Sobolev imbedding theorem, we have . It follows that in as for in as . This gives , i.e., for all . □

Let and , we denote

Thanks to Lemma 2.3, for some . We claim that for some . Indeed, if not, then as without loss of generality. On the one hand, since is a solution of (1.1), , where . On the other hand, by [[17], Lemma 3.7], we have

for and

for . Note that by a similar proof of [[14], Lemma 3.3], we can see that for μ small enough. Thus, for k large enough. This implies

for k large enough, which contradicts . For the sake of convenience, we denote , , by , , . Note that for every , is compact in (cf. [[9], Theorem 1.2]). It follows from [[19], Proposition 7.5] that there exists such that

(2.5)

Let and . Let . For small enough, we define , then we have the following.

Lemma 2.4Assume that there existssuch thatfornlarge. Then there existssuch thatforand largen.

Proof Assume a contradiction. Then, for every , there exists such that . It is clear that satisfies the (PS) condition for every . Hence there exists such that, up to a subsequence, in as with and . This implies

Thus, by [[9], Theorem 1.2], up to a subsequence, we see that there exists such that in as . Moreover, by using the arguments in the proof of Lemma 2.3, we have and . On the other hand, for large n, since . It follows that . This contradicts the fact that . □

Lemma 2.5Assume that there existssuch thatfor everyand largen. Then there existand an odd continuous mapsuch thatandfor largen.

Proof We first assume . It is clear that there exists such that

(2.6)

where

Let be the local Lipschitz continuous operator obtained in [[14], Lemma 2.1] and let be the solution of the following O.D.E.

Denote to be the maximal interval of existence of .

Claim 1: cannot enter before it enters for small δ, large n and .

Indeed, if the claim fails, then for every , will enter before it enters . Since , there exist such that for and

By [[14], Lemma 2.1], . On the other hand, by the choice of and , we know that for . Thanks to [[17], Lemma 3.8], for large n. This, together with (2.6) and [[14], Lemma 2.1], implies

Claim 2: There exists such that for large n and .

If the claim is not true, then for all . We first consider the case of . In fact, by Claim 1, , i.e., for all . Since for and large n, we must have

(2.7)

On the other hand, by [[14], Lemma 2.1] and [[17], Lemma 5.2], we have

This means for all , which contradicts with (2.7). It follows that there must exist such that for , large n and . Next, we consider the case of . Since for all and large n, it follows from [[14], Lemma 2.1] and [[17], Lemma 5.2] that

Thus, there also exists such that for and . Moreover, we must have for since for all .

Let

Then, by the continuity of , is continuous. Note that is odd and is even, we see that is odd and it is the desired map. The situation of can be proved in a similar way. Therefore, we complete the proof of this lemma. □

Proof of Theorem 1.1 We first consider the case . Thanks to Lemma 2.1 and [[9], Theorem 1.1], (1.1) has infinitely many sign-changing solutions. Next, we consider the case of . Since for every , for all , for all . It follows that two cases may occur:

Case 1: There are such that  .

In this case, Problem (1.1) has infinitely many sign-changing solutions.

Case 2: There exists such that for all .

In this case, if for every small enough, then Problem (1.1) also has infinitely many sign-changing solutions. Otherwise, there exists such that for . Thanks to Lemmas 2.4 and 2.5, there exists such that for small δ and large n. Fix and , the definitions of and give that there exists a large n such that and for small . By the definition of , there exists such that , where , and . It follows that . Thus, . By the choice of δ and , we have . If , then we have

A contradiction. By the properties of gen, we have

This implies . Since is arbitrary, we have , which contradicts with (2.5). □

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors typed, read and approved the final manuscript.

### Acknowledgements

The work was supported by the Natural Science Foundation of China (11071180, 11171247) and College Postgraduate Research and Innovation Project of Jiangsu Province (CXZZ110082).

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