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).
In this paper, we consider the following problem:
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.
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 [, 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.
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]
Proof By a contradiction, we may assume . By using a standard regularity argument and [, Lemmas 3.2 and 3.3], we have for some . Thus, it follows from the strong maximum principle (cf.) 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
is Fréchet differentiable. Let , where is a linearly independent sequence of . It is easy to show that there exists such that for , where (cf. [, Lemma 3.9]). We denote
By [, Theorem 1.2], we know that, for every , has infinitely many critical points, denoted by , in for μ small enough. Moreover,
Proof Consider the following auxiliary functional:
and the nonlinearity satisfies the assumptions of [, 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 [, Theorem 1.2], we know that for each , there exists such that as in . The next lemma will give more information about .
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
for n large enough. By (2.4), for this δ, we have
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 . □
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 [, Lemma 3.7], we have
for . Note that by a similar proof of [, 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. [, Theorem 1.2]). It follows from [, Proposition 7.5] that there exists such that
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 [, 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 . □
Let be the local Lipschitz continuous operator obtained in [, Lemma 2.1] and let be the solution of the following O.D.E.
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 [, Lemma 2.1] and [, Lemma 5.2] that
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:
In this case, Problem (1.1) has infinitely many sign-changing solutions.
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
The authors declare that they have no competing interests.
The authors typed, read and approved the final manuscript.
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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