Abstract and Applied Analysis

1. Introduction

In this paper, we deal with the multiplicity of positive solutions for the following semilinear elliptic equation: where , ( if , if ) and are measurable functions and satisfy the following conditions: , where , and there exist and such that , , is bounded and has a compact support in . There exist , and such that

Semilinear elliptic equations with concave-convex nonlinearities in bounded domains are widely studied. For example, Ambrosetti et al. [1] considered the following equation: where , . They proved that there exists such that () admits at least two positive solutions for all , has one positive solution for and no positive solution for . Actually, Adimurthi et al. [2], Damascelli et al. [3], Korman [4], Ouyang and Shi [5], and Tang [6] proved that there exists such that () in the unit ball has exactly two positive solutions for , has exactly one positive solution for and no positive solution exists for . For more general results of () (involving sign-changing weights) in bounded domains; see, the work of Ambrosetti et al. in [7], of Garcia Azorero et al. in [8], of Brown and Wu in [9], of Brown and Zhang in [10], of Cao and Zhong in [11], of de Figueiredo et al. in [12], and their references.

However, little has been done for this type of problem in . We are only aware of the works [13–17] which studied the existence of solutions for some related concave-convex elliptic problems (not involving sign-changing weights). Furthermore, we do not know of any results for concave-convex elliptic problems involving sign-changing weight functions except [18, 19]. Wu in [18] have studied the multiplicity of positive solutions for the following equation involving sign-changing weights: where the parameters . He also assumed that is sign chaning and , where and satisfy suitable conditions and proved that () has at least four positive solutions.

In a recent work [19], Hsu and Lin have studied () in with a sign-changing weight function. They proved there exists such that () has at least two positive solutions for all provided that , satisfy suitable conditions and maybe changes sign in .

Continuing our previous work [19], we consider () in involving a sign-changing weight function with suitable assumptions which are different from the assumptions in [19].

In order to describe our main result, we need to define where , and is the best Sobolev constant for the imbedding of into .

Theorem 1.1. Assume that , - hold. If , () admits at least two positive solutions in .

This paper is organized as follows. In Section 2, we give some notations and preliminary results. In Section 3, we establish the existence of a local minimum. In Section 4, we prove the existence of a second solution of ().

At the end of this section, we explain some notations employed. In the following discussions, we will consider with the norm . We denote by the best constant which is given by The dual space of will be denoted by . denote the dual pair between and . We denote the norm in by for . is a ball in centered at with radius . denotes as . , will denote various positive constants, the exact values of which are not important.

2. Preliminary Results

Associated with (1.3), the energy functional defined by for all is considered. It is well-known that and the solutions of () are the critical points of .

Since is not bounded from below on , we will work on the Nehari manifold. For we define Note that contains all nonzero solutions of () and if and only if

Lemma 2.1. is coercive and bounded from below on .

Proof. If , then by , (2.3), and the Hölder and Sobolev inequalities, one has Since , it follows that is coercive and bounded from below on .

The Nehari manifold is closely linked to the behavior of the function of the form for . Such maps are known as fibering maps and were introduced by Drábek and Pohozaev in [20] and are also discussed by Brown and Zhang in [10]. If , we have It is easy to see that and so, for and , if and only if that is, the critical points of correspond to the points on the Nehari manifold. In particular, if and only if . Thus, it is natural to split into three parts corresponding to local minima, local maxima, and points of inflection. Accordingly, we define and note that if , that is, , then

We now derive some basic properties of , , and .

Lemma 2.2. Suppose that is a local minimizer for on and , then in .

Proof. See the work of Brown and Zhang in [10, Theorem 2.3].

Lemma 2.3. If , then .

Proof. We argue by contradiction. Suppose that there exists such that . Then for by (2.9) and the Sobolev inequality, we have and so Similarly, using (2.10), Hölder and Sobolev inequalities, we have which implies Hence, we must have which is a contradiction.

In order to get a better understanding of the Nehari manifold and fibering maps, we consider the function defined by Clearly, if and only if . Moreover, and so it is easy to see that if , then . Hence, (or ) if and only if (or ).

Let . Then, by (2.17), has a unique critical point at , where and clearly is strictly increasing on and strictly decreasing on with . Moreover, if , then

Therefore, we have the following lemma.

Lemma 2.4. Let and . (i)If , then there exists a unique such that , is inceasing on and decreasing on . Moreover, (ii)If , then there exist unique such that , , is decreasing on , inceasing on and decreasing on (iii).(iv)There exists a continuous bijection between and . In particular, is a continuous function for .

Proof. See the work of Hsu and Lin in [19, Lemma 2.5].

We remark that it follows Lemma 2.4, for all . Furthermore, by Lemma 2.4 it follows that and are non-empty and by Lemma 2.1 we may define

Theorem 2.5. (i) If , then we have .
(ii) If , then for some .
In particular, for each , we have .

Proof. See the work of Hsu and Lin in [19, Theorem 3.1].

Remark 2.6. (i) If , then by (2.9), Hölder and Sobolev inequalities, for each we have and so
 (ii) If , then by Lemma 2.4(i), (ii) and Theorem 2.5(ii), for each we have

3. Existence of a Positive Solution

First, we define the Palais-Smale (simply by ) sequences, -values, and -conditions in for as follows.

Definition 3.1. (i) For , a sequence is a -sequence in for if and strongly in as .
(ii) is a -value in for if there exists a -sequence in for .
(iii) satisfies the -condition in if any -sequence in for contains a convergent subsequence.

Now we will ensure that there are -sequence and -sequencein on and , respectively, for the functional .

Proposition 3.2. If , then (i)there exists a -sequence in for .(ii)there exists a -sequence in for .

Proof. See Wu [21, Proposition 9].

Now, we establish the existence of a local minimum for on .

Theorem 3.3. Assume and hold. If , then there exists such that (i), (ii) is a positive solution of (), (iii) as .

Proof. From Proposition 3.2(i) it follows that there exists satisfying By Lemma 2.1 we infer that is bounded on . Passing to a subsequence (Still denoted by ), there exists such that as By , Egorov theorem and Hölder inequality, we have By (3.1) and (3.2), it is easy to see that is a solution of (). From and (2.4), we deduce that Let in (3.4). By (3.1), (3.3) and , we get Thus, is a nonzero solution of ().
Next, we prove that strongly in and . From the fact and applying Fatou's lemma, we get This implies that and . Standard argument shows that strongly in . By Theorem 2.5, for all we have that and which implies . Since and , by Lemma 2.2 we may assume that is a nonzero nonnegative solution of (). By Harnack inequality [22] we deduce that in . Finally, by (2.10), Hölder and Sobolev inequlities, and thus we conclude the proof.

4. Second Positive Solution

In this section, we will establish the existence of the second positive solution of () by proving that satisfies the -condition.

Lemma 4.1. Assume that and hold. If is a -sequence for , then is bounded in .

Proof. See the work of Hsu and Lin in [19, Lemma 4.1].

Let us introduce the problem at infinity associated with (): We state some known results for problem (). First of all, we recall that by Lions [23] has studied the following minimization problem closely related to problem (): where . Note that a minimum exists and is attained by a ground state in such that where . Gidas et al. [24] showed that for every , there exist positive constants , such that for all , We define Clearly, .

Lemma 4.2. Let be a domain in . If satisfies then

Proof. We know . Then, Since as , then the lemma follows from the Lebesgue dominated convergence theorem.

Lemma 4.3. Under the assumptions , - and . Then there exists a number such that for In particular, for all .

Proof. (i) First, since for all and is continuous in and , we infer that there exists such that
 (ii) Since , there exists such that if , we get for . Then, for Thus, there exists such that for any and we get
 (iii) By (i) and (ii), we need to show that there exists such that for We know that . Then, , we have Suppose satisfies , we get for all and some positive constant . By (4.3) and Lemma 4.3, there exists such that for any By and (4.3), we get By , (4.3) and Lemma 4.3, we have Since and and using (4.13)–(4.16), we have there exists such that for all , then This implies that if , then for all we get From for all and (4.17), we have Combining this with Lemma 2.4(ii), from the definition of and , for all , we obtain that there exists such that and