Research Article | Open Access
The Choquard Equation with Weighted Terms and Sobolev-Hardy Exponent
We study a nonlinear Choquard equation with weighted terms and critical Sobolev-Hardy exponent. We apply variational methods and Lusternik-Schnirelmann category to prove the multiple positive solutions for this problem.
The main goal of this paper is to consider the multiplicity of positive solution for the following nonlinear Choquard equation:where is a bounded open set with smooth boundary. , , , , and is a parameter. There are two continuous weight functions, satisfying the following conditions: . and . there exist two positive constants such that and for every . , , there exists such that
In recent years, much attention has been paid to nonlinear Choquard equation involving the Hardy-Littlewood-Sobolev inequality, which generalizes and complements the classical elliptic boundary value problems and Schrödinger-Poisson system [1–21]. In particularly, more and more authors have studied the critical problems [22–36]. In the case that , Gao and Yang  considered the existence and multiplicity of (1) with upper critical exponent . The authors focus on the case that how the sublinear perturbation term has influence on the multiplicity of (1). When , Xiang  showed the uniqueness and nondegeneracy solutions for ground states of Choquard equation. Furthermore, by means of the tool of Nehari manifold, Zhang et al.  established the existence theorem of ground states for generalized Choquard equation when the nonlinear term is concave-convex. On the other hand, there are a great deal of results on the existence of elliptic boundary value problems with sign-changing weights [37–45]. We should point out that Hsu and Lin  showed the existence of positive solutions for elliptic equations with concave-convex nonlinearities and sign-changing weights. What is more, Wu  obtained three positive solutions discussed in  by using of the method of Nehari manifold combining with the Lusternik-Schnirelmann category. In view of the same method, Chen and Wu  obtained the existence of positive solutions for a class of critical semilinear problem. Chen et al.  established multiplicity theorems for Kirchhoff type equation with sign-changing weight functions. Goyal and Sreenadh  used fibering map approach to study -fractional Laplacian equation with sign-changing weight function.
Motivated by above results, we use variational approach to analyze the existence and multiplicity of positive solutions for (1). To the best of our knowledge, there is no result studying Choquard equation with upper critical exponent and two sign-changing weight functions.
Our main results are the following theorems.
Theorem 1. Assume that hold, for every , the problem (1) has at least one positive solution in .
Theorem 2. Assume that hold; there exists , such that, for every , the problem (1) has at least two positive solutions in .
Furthermore, we will utilize the following condition.
(G) There are a nonempty closed set and such that
Remark 3. Let for . According to (G), we can assume that there exist two positive solutions and such that
Theorem 4. Assume that holds; for every , there exists such that for , (1) has at least positive solutions.
Proposition 6 (Hardy-Littlewood-Sobolev inequality ). Let and with and . There exists a constant independent of , such that If , then For , by Proposition 6, we have The best constant is defined as
Proposition 7 (see ). The constant is attained if and only if where ; therefore whereFor is a solution of the problem with = .
Definition 8. (i) For , if strongly in as , then the sequence is a sequence in ;
(ii) satisfies the condition if every sequence in for has a convergent subsequence.
Though is not bounded below on , we can construct the following Nehari manifold: Define the fiber map by .
Lemma 9. is coercive and bounded below on .
Proof. If , by (14) we haveso is coercive and bounded below on . This lemma is completed.
Lemma 10. Assume that is a local minimizer for on and . Then . Moreover, if is a nontrivial nonnegative function in , then is a positive solution of (1).
Proof. Since is a local minimizer for on , is a solution of the optimization problem where Hence, using Lagrange multipliers, there exists , such that , which implies Since and , one has Hence, if , combining (21) , then in . Using Harnack inequality, thus, is a positive solution of (1) in . The lemma is completed.
From , we obtain that for , we get Obviously, ; we divide into the following three sets
For any , one hasTherefore, we can derive
Lemma 11. (i) For every , then .
(ii) For every , then .
Lemma 12. For , then .
Proof. Assuming the contrary, there is such that , then for , by (26) and Proposition 6, which implies so Similarly, coupling with (26) and (14), which infers that hence From the above inequalities, we deduce that which is a contradiction.
Letting , one has Let Assume if and only if So Clearly, , then . Hence, (or ) provided (or . can attain its maximum at duo to , where where
Lemma 13. For every , it follows that
(i)if , there exists a unique such that , and is increasing on , decreasing on .
Moreover (ii)if , then there exist unique such that and is increasing on , decreasing on . Moreover
Proof. (i)Assume that . There exists a solution , with , then , and . Consequently, has a unique critical point at and . Since . Hence, (36) holds and .
(ii) Assume , then . There are two solutions , satisfying , such that , which means that these two solutions of depend on , where . So according to has critical points at . Thus, is increasing on , and , (37) holds.
This lemma is completed.
Furthermore, we have Define
Theorem 14. (i) for all .
(ii) If , then there exists a positive number , such that .
Especially, for all .
Lemma 15. Every sequence that satisfies with and = has a convergent subsequence.
Proof. For , there exists such thatTherefore So, by Theorem 14, , we know that Thus which implies Furthermore
Lemma 16. For every , then has a minimizer in , satisfying that(i);(ii) is a positive solution of (1);(iii) as