Quasilinear Elliptic Equations with Hardy-Sobolev Critical Exponents: Existence and Multiplicity of Nontrivial Solutions
We study the existence of positive solutions and multiplicity of nontrivial solutions for a class of quasilinear elliptic equations by using variational methods. Our obtained results extend some existing ones.
1. Introduction and Main Results
Let us consider the following problem: where denotes the -Laplacian differential operator, is an open bounded domain in with smooth boundary and , , is the Hardy-Sobolev critical exponent, and is the Sobolev critical exponent. Here, we let which is equivalent to the usual norm of Sobolev space due to the Poincaré inequality. Let which is the best Hardy-Sobolev constant.
In the case where and hold, then (1) reduces to the quasilinear elliptic problem: Gonçalves and Alves  have studied (4) in involving , and to obtain existence of positive solutions where , , or and a suitable . We should mention that problem (4) with has been widely studied since Brézis and Nirenberg; see [2–4] and the references therein.
Ghoussoub and Yuan  have studied (1) with , where . They obtained a positive solution in the case where , , and (in particular if ) hold. They also obtained a sign-changing solution in the case where , , and (in particular if ) hold. For other relevant papers, see [6–12] and the references therein.
We should mention that the energy functional associated with (1) is defined on , which is not a Hilbert space for . Due to the lack of compactness of the embedding in and , we cannot use the standard variational argument directly. The corresponding energy functional fails to satisfy the classical Palais-Smale ( for short) condition in . However, a local () condition can be established in a suitable range. Then the existence result is obtained via constructing a minimax level within this range and the Mountain Pass Lemma due to Ambrosetti et al.  and Rabinowitz .
In this paper, we study (1) with a general nonlinearity by using a variational method; besides, we also considerably generalize the results obtained in . In what follows, we always assume that the nonlinearity satisfies . Let , . To state our main results, we still need the following assumptions., , and uniformly for .There exists a constant with such that , , and uniformly for .There exists a constant with such that
Now, our main results read as follows.
Theorem 1. Suppose that , , , and hold. If , , and hold, then (1) has at least one positive solution.
Noting that and imply that (7) holds, therefore, we have the following corollaries.
Corollary 3. Suppose that , , , and . Moreover, and hold; then (1) has at least one positive solution.
Corollary 4. Suppose that , , , and . Moreover, and hold; then (1) has at least two distinct nontrivial solutions.
Remark 5. Theorem 1 generalizes Theorem 1.3 in , where the author only studied the special situation that , . There are functions satisfying the assumptions of our Theorem 1 and not satisfying those in . Let where , , , and . Obviously, satisfies all the conditions of Theorem 1 in this paper, while it does not satisfy the conditions of Theorem 1.3 in .
The rest of this paper is organized as follows. In Section 2, we give some preliminary lemmas, which are useful in the proofs of our main results. In Section 3, we give the detailed proofs of our main results.
In what follows, we let denote the norm in . It is obvious that the values of for are irrelevant in Theorem 1, so we may define We firstly consider the existence of nontrivial solutions to the problem: The energy functional corresponding to (10) is given by By Hardy-Sobolev inequalities (see [5, 14]) and , we know . Now it is well known that there exists a one-to-one correspondence between the weak solutions of (10) and the critical points of on . More precisely we say that is a weak solution of (10), if, for any , there holds
Lemma 6 (see ). If a.e. in and for all and some , then
Lemma 7. For any , , and , we have .
Proof. Let Clearly, , , so .
Lemma 8 (see ). If hold, then we have(i) is independent of , and will henceforth be denoted by ;(ii) is attained when by the functions for some . Moreover, the functions are the only positive radial solutions of in , and satisfy
Lemma 9. If , , and hold, then satisfies condition.
Proof. Suppose that is a sequence in . By , we have where . Hence we conclude that is a bounded sequence in . So there exists ; going if necessary to a subsequence, we have By the continuity of embedding, we have . From , going if necessary to a subsequence, one can get that as . By , we know that for any there exists such that Set . When , , we get It follows from Vitali’s theorem that Similarly, we can also get Since , we have Let , which together with Lemma 6 implies From (20), we can obtain Note that as , which together with Lemma 6 implies Therefore, one gets that From (26) and (27), we have then as . Otherwise, there exists a subsequence (still denoted by ) such that By (3), we have then . That is, . It follows from (29) and that . However, we have by (27) and (). We get a contradiction. Therefore, we can obtain From the discussion above, satisfies condition.
In the following, we shall give some estimates for the extremal functions. Let Define a function such that for , for , , where . Set so that . Then, by using the argument as used in , we can get the following results: Moreover, by using the Sobolev embedding theorem and (36), one can deduce
Lemma 10. Suppose that . If , and (7) hold, then there exists , , such that
Proof. We consider the functions Since , , and for small enough, is attained for some . Therefore, we have and hence Therefore, we obtain By (), we can easily get Hence, we can get By (36)–(38), when is small enough, we conclude that On the one hand, from Lemma 7 and (36), it follows that On the other hand, the function attains its maximum at and is increasing in the interval . Note that () implies , which together with (36), (46), and (47) implies that Furthermore, from (7) and (37), we get By (7), we have , which implies Therefore, by choosing small enough, we have Hence, the proof of the lemma is completed by taking .
3. Proofs of Main Results
Proof of Theorem 1. Let . From the Sobolev and Hardy-Sobolev inequalities, we can easily get It follows from () that uniformly for all . Therefore, we deduce that for all and for . Then one gets for all and for . By (52) and (55) we have for small enough. So there exists such that By Lemma 10, there exists with such that It follows from the nonnegativity of that Therefore, , so we can choose such that By virtue of the Mountain Pass Lemma in , there is a sequence satisfying where Note that By Lemma 9 we can assume that in . From the continuity of , we know that is a weak solution of problem (10). Then , where . Thus . Therefore, is a nonnegative solution of (1). By the Strong Maximum Principle , is a positive solution of problem (1). Therefore, Theorem 1 holds.
Proof of Theorem 2. By Theorem 1, we know that (1) has a positive solution . Set It follows from Theorem 1 that the equation has at least one positive solution . Let ; then is a solution of It is obvious that , , and . So (1) has at least two nontrivial solutions. Therefore, Theorem 2 holds.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author thanks the referees and the editors for their helpful comments and suggestions. Research is supported by the Tianyuan Fund for Mathematics of NSFC (Grant no. 11326113) and the Key Project of Natural Science Foundation of Educational Committee of Henan Province of China (Grant no. 13A110015).
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