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Journal of Function Spaces
Volume 2018, Article ID 6108538, 14 pages
https://doi.org/10.1155/2018/6108538
Research Article

Twin Positive Solutions for Schrödinger-Kirchhoff-Type Problem with Singularity and Critical Exponents

Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China

Correspondence should be addressed to Wei Han; moc.621@1iewiewnah_hs

Received 27 January 2018; Accepted 25 March 2018; Published 2 May 2018

Academic Editor: Dumitru Motreanu

Copyright © 2018 Wei Han and Yangyang Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study in this paper the following singular Schrödinger-Kirchhoff-type problem with critical exponent in on where are constants, is a smooth bounded domain, , is a real parameter, is a constant, and ( is the first eigenvalue of , under Dirichlet boundary condition). Under appropriate assumptions on and , we obtain two positive solutions via the variational and perturbation methods.

1. Introduction and Main Result

In this paper, we consider the following Schrödinger-Kirchhoff-type problem with Dirichlet boundary value conditionswhere is a smooth bounded domain in , , is a real parameter, is a constant, and ( is the first eigenvalue of , under Dirichlet boundary condition), and coefficient functions , are positive, which have at least one minimum. This kind of problem has been widely concerned. The Schrödinger-Kirchhoff-type problem in the effect of electromagnetic field on current motion in physics and the basic law of physical world material movement has been widely used. In recent years, the following elliptic problems have been extensively studied by many researchers:where , , and are constants. In [1], Li and Ye considered the existence of positive ground state solutions to the following Kirchhoff-type problem with pure power nonlinearities: Furthermore, Chen and Tang [2] studied the following Kirchhoff-type equation: and their result unifies both asymptotically cubic and supercubic cases, which generalizes and improves the existing ones. In [3], Huy and Quan investigated the equation in , on . They proved the existence results for both nondegenerate and degenerate cases of the function and combined the fixed point index theory with the cone theoretic argument which are key technical ingredients to obtain the main results. In [4], Figueiredo and Severo established the existence of a positive ground state solution for a Kirchhoff problem in involving critical exponential growth. Moreover, Ricceri [5] obtained some results of a totally new type about a class of nonlocal problems. In [6], the authors studied the multiplicity of positive solutions for a class of Kirchhoff type of equations with the nonlinearity containing both singularity and critical exponents:In the case that in (2), it is related to the stationary analogue of the equation presented by Kirchhoff in [7]. In [8], Lions introduced an abstract functional analysis framework to the following equation:proposed by He and Zou in [9] as an existence of the classical D’Alembert’s wave equations for free vibration of elastic strings. After the vanguard work of Lions [8], where a functional analysis method was presented, the Kirchhoff-type equations started to arouse attention of researchers. D’Ancona and Spagnolo [10] studied the existence of a global classical periodic solution with real analytic data. Arosio and Panizzi [11] proved the well-posedness of the Cauchy problem concerned (7) in the Hadamard sense as a special case of an abstract second-order Cauchy problem in a Hilbert space. Perera and Zhang [12] obtained a nontrivial solution via Yang index and critical group. Caponi and Pucci [13] proved existence, multiplicity, and asymptotic behavior of entire solutions for a series of stationary Kirchhoff fractional -Laplacian equations. In [1416], the authors proved the existence of a nontrivial solution without the Ambrosetti-Rabinowitz condition. Moreover, we found in [1719] that the authors considered the existence and multiplicity of solutions by Nehari manifold. In [20], Shen and Yao obtained two positive solutions about the following Kirchhoff problem:under some assumptions on the weight functions and . The related developments of the existence of solutions for the Kirchhoff-type problems also can be found in [2124].

However, up till now, no paper has appeared in the literature which discusses the singular Schrödinger-Kirchhoff-type problem with critical exponent. This paper attempts to fill in this gap in the literature; in this paper, we obtain the multiple positive solutions for the problem via the variational and perturbation methods. Throughout this paper, we make use of the following notations:

(1) The norm in is equipped with the norm , and the norm in is denoted by .

(2) , .

(3) , , and denote various positive constants, which may vary from line to line.

(4) We denote by (resp., ) the sphere (resp., the closed ball) of center zero and radius ; that is, , . Let be the Sobolev constant; namely,We define the functionalIn general, a function is called a weak solution of (1) if , and for all , it holds that and for , we consider the following perturbed problem:Solution (12) corresponds to critical point of -functional on by The main result can be described as follows.

Theorem 1. Assume that , , , and , are positive, which have at least one minimum. Then, there exists , such that, for any , problem (1) has at least two different positive solutions.

This paper is organized as follows. In Sections 2 and 4, we give the proof of Theorem 1. In Section 3, we consider the existence of at least one mountain-pass solution of the perturbation problem.

2. Existence of a First Solution of Problem (1)

In this section, our main work is to prove that problem (1) has a local minimum solution in .

Lemma 2. There exist constants and , such that, for all , one has

Proof. By Hölder inequality and (9), we can obtain thatTherefore, from (9) and (15), we haveWe can set for ; therefore, there exists a constant , such that . We can let ; then, there exists a constant such that for all . For given , we can deduce with ; one has Hence, we have for all and small enough. So, for sufficiently small, we haveThis completes the proof of Lemma 2.

Theorem 3. Assume , where is defined in Lemma 2. Hence, one obtains that problem (1) has a positive solution , with .

Proof. First, there exists , such that . Actually, from (14), we can obtain that With the definition of , we have that there exists a minimizing sequence such that . Therefore, is bounded in . Hence, there exists a subsequence, still denoted by . We assume that there exists such thatFor , by Hölder inequality, one hasTherefore, by (21), we can obtain thatLet ; by Brézis Lemma, we getIf we set , it follows that and . If , by (24), one has for large enough. Therefore, by (20), we haveHence, noticing (23)–(28), we get Passing to the limit as , we have . Noticing that is closed and convex, so . Therefore, from (18), we get and . Thus, it can be seen that is a local minimizer of .
Next, we prove that is a solution of (1) and . By the above expression, we can obtain that is a local minimizer of . So, for any , , set sufficiently small, such that ; we haveCurrently, we shall prove that is a solution of (1). Indeed, by (30), one has Dividing by and taking the limit , one getsNote thatwhere and a.e. as , for . Therefore, using Fatou Lemma, one has Hence, by (32) and the above estimate, we can obtain thatFor and (14), we can get ; then, . For , there is such that for . Define by . Actually, when , achieves its minimum; namely,Assume and . Define by . So, from (35) and (36), we can getBy the measure of the domain of integration as , it can be seen that Then, dividing by and letting in (37), one hasWith the arbitrariness of , this inequality also holds for ; that is,On the one hand, we can take the test function in (40); we have , and then we get . Therefore, is a nonzero solution of (1).
On the other hand, let be a family of positive solutions of (12); with an easy computation, we see that where , , which means that and satisfies Noticing , , thus, we have , in by the strong maximum principle.
From the above discussion, we can deduce that is a positive solution of (1) with . This completes the proof of Theorem 3.

3. Existence of a Mountain-Pass Solution of Problem (12)

In this section, we consider the existence of at least one mountain-pass solution of the perturbation problem.

Lemma 4. Suppose that . For ,, where , , and are as in Lemma 2, the functional satisfies the following conditions:(i), if .(ii)There exists such that , and .

Proof. (i) Notice the subadditivity of ; namely,From (43), we can get Hence, when , from (14), we can get the conclusion.
(ii) For every , , and , one has as . Hence, we can consider such that and . This completes the proof of Lemma 4.

We define

Lemma 5. Let ; then satisfies the condition for every , where , and is as in (9).

Proof. Let , soWe shall claim that is bounded in . Otherwise, we can suppose that , as . From (46), (43), and (15), one has By , it can be seen that the last inequality is unreasonable. Clearly, is bounded in . Up to a subsequence, there exists such thatSince and , using the dominated convergence theorem and (48), we getFurthermore, from and (48), we getHence, it follows from the dominated convergence theorem thatLet ; then, as . Otherwise, there exists a subsequence (still denoted by ) such that (where is a positive constant). Therefore, by and (49), one has