Infinitely many sign-changing solutions for p-Laplacian equation involving the critical Sobolev exponent
© Wu and Huang; licensee Springer. 2013
Received: 14 December 2012
Accepted: 28 May 2013
Published: 19 June 2013
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).
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 . 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  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 , 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  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  have shown that (1.1) with has infinitely many sign-changing solutions when and , which extends the main result in .
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 , 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 1.1 Assume that and . Then Problem (1.1) has infinitely many sign-changing solutions.
where and is increasing to . It has been shown by [, 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 [, 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. ). Moreover, by [, 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 [, Lemma A.6]
and the equality holds if and only if for some constant .
Lemma 2.1 Assume that is a nonzero solution of (1.1) for . Then u is sign-changing.
which is impossible since , , and . Therefore, we have proved Lemma 2.1. □
Next, we consider the case of .
Here, we say that A is symmetric if implies .
Lemma 2.2 For every , there exists such that for all .
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 [, Theorem 1.2], we know that for each , there exists such that as in . The next lemma will give more information about .
Lemma 2.3 is a sign-changing solution of Problem (1.1) for every .
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 . Let . For small enough, we define , then we have the following.
Lemma 2.4 Assume that there exists such that for n large. Then there exists such that for and large n.
Thus, by [, 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 2.5 Assume that there exists such that for every and large n. Then there exist and an odd continuous map such that and for large n.
Denote to be the maximal interval of existence of .
Claim 1: cannot enter before it enters for small δ, large n and .
A contradiction with .
Claim 2: There exists such that for large n and .
Thus, there also exists such that for and . Moreover, we must have for since for all .
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 [, 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 .
This implies . Since is arbitrary, we have , which contradicts with (2.5). □
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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