#### Abstract

Existence and multiplicity of positive solutions for the following semilinear elliptic equation: in , , are established, where if if , , satisfy suitable conditions, and maybe changes sign in . The study is based on the extraction of the Palais-Smale sequences in the Nehari manifold.

#### 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

Lemma 4.4. *Assume that and hold. If is a -sequence for with , then there exists a subsequence of converging weakly to a nonzero solution of () in .*

* Proof. *Let be a -sequence for with . We know from Lemma 4.1 that is bounded in , and then there exist a subsequence of (still denoted by ) and such that
It is easy to see that and by , Egorov theorem and Hölder inequality, we have

Next we verify that . Arguing by contradiction, we assume . By , for any , there exists such that for all . Since strongly in for , is a bounded sequence in , therefore . Setting , then , we have
We set
Since and is bounded, then by (4.22), we can deduce that
that is,

If , then we get , which contradicts to . Thus we conclude that . Furthermore, by the definition of we obtain
Then, as , we have
which implies that
Hence, from (4.2) and (4.22)–(4.29), we get
This is a contradiction to . Therefore, is a nonzero solution of ().

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

Theorem 4.5. *Assume that and - hold. If , then there exists such that *(i)*,*(ii)* is a positive solution of (). *

* Proof. *If , then by Theorem 2.5(ii), Proposition 3.2(ii) and Lemma 4.3(ii), there exists a -sequence in for with . From Lemma 4.4, there exist a subsequence still denoted by and a nonzero solution of () such that weakly in .

First, we prove that . On the contrary, if , then by is closed in , we have . From (2.9) and for all , we get
By Lemma 2.4(ii), there exists a unique such that . If , then it is easy to see that
From (3.1), and (4.32), we can deduce that
which is a contradiction. Thus, .

Next, by the same argument as that in Theorem 3.3, we get that strongly in and for all . Since and , by Lemma 2.2 we may assume that is a nonzero nonnegative solution of (). Finally, by the Harnack inequality [22] we deduce that in .

Now, we complete the proof of Theorem 1.1. By Theorems 3.3, 4.5, we obtain () has two positive solutions and such that , . Since , this implies that and are distinct. It completes the proof of Theorem 1.1.