Existence and Multiplicity of Positive Solutions for Schrödinger-Kirchhoff-Poisson System with Singularity
We study a singular Schrödinger-Kirchhoff-Poisson system by the variational methods and the Nehari manifold and we prove the existence, uniqueness, and multiplicity of positive solutions of the problem under different conditions.
In the present paper, we consider the following singular Schrödinger-Kirchhoff-Poisson system: where is a smooth bounded domain with boundary , , , , and are four parameters and , is a constant, and is a nontrivial nonnegative function.
When , , , and , system (1) reduces to the following singular Schrödinger-Poisson system: which has been studied in . By using the variational method and the Nehari manifold, the existence, uniqueness, and multiplicity of solutions for system (2) have been obtained.
When , the system (1) reduces to the following singular Kirchhoff type problem:
It is well known that the following Kichhoff type problem: where is a smooth bounded domain and is a continuous function, has been extensively studied; see [4–7] and so forth. Problem (5) is called nonlocal because of the presence of the term which implies, when , that the equation in (5) is no longer a pointwise identity.
Recently, the singular Kirchhoff type problems have been considered (see [8–11]). In , by using the Nehari manifold methods, Liu and Sun proved that problem (5) with has at least two positive solutions for small enough. In , by using the variational methods, Lei et al. obtained that problem (5) with has at least two positive solutions for small enough. The common characteristic of [8–10] is that the nonlinear terms contain both singular and superlinear. The superlinear term can overcome the difficulties which are caused by the Kirchhoff type perturbation .
Motivated by the above references, especially by [8–10], we study the singular Schrödinger-Kirchhoff-Poisson system (1). Let be the Sobolev space equipped with the inner product and norm As usual, , , denotes the norm of the Lebesgue space . Let be the usual Sobolev constant defined by that is, By the Hölder inequality and (8), we havewhere denotes the Lebesgue measure of the domain .
Lemma 1. Let ; then for every , there exists a unique solution of Moreover(i);(ii) Moreover when and ;(iii)for each , ;(iv)(v)assume that ; then in and for any ;(vi);(vii)for , .
As far as we know, the singular Schrödinger-Kirchhoff-Poisson system has not been considered up to now, and the study on the existence, uniqueness, and multiplicity of solutions for system (1) is meaningful in mathematics. We emphasize that the combined effects of the two nonlocal terms it contains cause some mathematical difficulties which make the study of such class of problems particularly interesting.
We consider system (1) in two cases: and . By using the variational method, the Nehari manifold, and Ekeland’s variational principle, we obtain the existence and uniqueness of positive solution for and the multiplicity of positive solutions for . To prove the multiplicity of positive solutions, it is necessary that , where are defined in Section 3.2. But the method of getting in  cannot apply to our problem. In order to show that , we introduce a special nonempty set. Let Our main results can be described as follows.
Theorem 2. Assume , , , and , then system (1) has a unique positive solution for all and .
2. Proof of Theorem 2
Lemma 5. For all , , and , the functional attains the global minimizer in ; that is, there exists such that
Proof of Lemma 5. For , by Lemma 1(ii) and (9),so, is coercive and bounded from below on for any . Thus is well defined. For and given ,we can see that for small enough, . Therefore, we have .
According to the definition of , there exists a minimizing sequence such that . Since , we may assume that From (18), it is easy to see that is bounded in , up to a subsequence; there exists such thatThen by the weakly lower semicontinuity of the norm, Lemma 1(v) and (14), we haveSo, we have
Proof of Theorem 2. We divide three steps to prove Theorem 2.
(1) We show in .
From Lemma 5, and . Fix , , and ; we havethat isNotice thatwhere and ; a.e , . Since , by using Fatou’s lemma, we haveBy the idea of approximation, the above expression also holds for , ; that is,Therefore,Since and , by Lemma 1(ii) and (vi), and . Therefore, by the strong maximum principle for weak solutions, we obtain that a.e. in
(2) We show that is a solution of (12); that is, we prove satisfying (15) for .
For given , define by ; then attains its minimum at by Lemma 5. It implies thatWe take , and define . LetThen , . Inserting into (26) and using (28), we can obtain thatSince and the measure of the domain tends to zero as , it follows thatThen dividing by and letting in (30), we see thatThis inequality also holds for , so we getThen is a solution of (12) for , , and .
(3) We show that is the unique solution of (12) for , , and .
Assume that is also a solution of (12) for , , and . It follows from (15) that Subtracting (34) and (35), we obtain thatSince , in , the following inequality holds:Consequently, it follows from Lemma 1(vii) and (36) thatwhich implies thatthat is is the unique solution of system (1).
3. Proof of Theorem 3
3.1. The Case of
In this part, let , where From Lemma 1(iv) and (9), we haveThus, is coercive and bounded from below on for all and . Sois well defined. Obviously, . Similar to the proof of Lemma 5, we easily obtain the following proposition.
Proposition 6. For all , , , and , the functional attains the global minimizer in ; that is, there exists such that
3.2. The Case of
In this part, let . is not necessarily coercive and bounded from below on for . We define a Nehari manifold by
Then the solutions of (12) must lie in . Obviously, is a closed set in . In order to obtain the multiplicity, we split into the following three parts:When , we have
Lemma 7. For any there exists such that and for .
Proof. For any given , , by calculating, we can get thatLetThen,Since , it is easy to obtain thatTherefore, for any , let . ThenMoreover, for all and for all . Thus is increasing for all and decreasing for all . SoThus from Lemma 1(iv), (9), and (53), it follows thatfor all .
From (54) and (55), we obtain that there exist unique positive numbers such thatFrom (48)–(50), it follows that and for all .
Next, we prove that for all . Assume that there exists and . Then it follows thatFrom (57) and (55), one has andfor all , which implies a contradiction. Thus for all .
Lemma 8. is a closed set in for .
Proof. Suppose such that in as . From the definition of , one hasThus it follows from in as thatso . If , then . However, from (60), we obtainwhich contracts . Thus for all . Therefore, is closed for all .
Lemma 9. Assume and , is coercive and bounded from below on and .
Proof. For all , from (9) we haveIt follows from that is coercive and bounded from below on for any .
Since and are two nonempty closed subsets of for all and , it follows that and are well defined. For any given , by (47), we haveConsequently,Thus, by Lemma 7, for all and
Lemma 10. Assume that and , given (resp., ), there exists and a continuous function , , , satisfying that
Proof. Please see Lemma of  for the similar proof.
Proof of Theorem 3(ii). By Ekeland’s variational principle, there exists a minimizing sequence satisfying(i),(ii), From and Lemma 9, we may assume that and It is easy to obtain that is bounded in ; we assume that . Going if necessary to a subsequence, we can assume thatSince , we getBy the weakly lower semicontinuity of the norm, Lemma 1(v), (14), and (47), we haveTherefore, and .
First, we prove that in Since , we can claim that there exists a constant such that up to a subsequence we haveIn order to prove (70), it suffices to verifySince , It follows thatBy contradiction, we assume thatFrom the boundedness of and (72), one haswhich combines with (74); it follows thatLet ; then andSince , one hasConsequently, we haveFor all , from (55), (77), and (79),