Abstract and Applied Analysis

Abstract and Applied Analysis / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 658397 | https://doi.org/10.1155/2010/658397

Tsing-San Hsu, Huei-Li Lin, "Multiple Positive Solutions for Semilinear Elliptic Equations in ℝ 𝑁 Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions", Abstract and Applied Analysis, vol. 2010, Article ID 658397, 21 pages, 2010. https://doi.org/10.1155/2010/658397

Multiple Positive Solutions for Semilinear Elliptic Equations in ℝ 𝑁 Involving Concave-Convex Nonlinearities and Sign-Changing Weight Functions

Academic Editor: Norimichi Hirano
Received01 Nov 2009
Revised19 Mar 2010
Accepted09 Jun 2010
Published19 Jul 2010

Abstract

We study the existence and multiplicity of positive solutions for the following semilinear elliptic equation in , , where , if , if ), , satisfy suitable conditions, and may change sign in .

1. Introduction and Main Results

In this paper, we deal with the existence and multiplicity of positive solutions for the following semilinear elliptic equation: where , ( if , if ), and are measurable functions and satisfy the following conditions: with in ; and in .

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 and has one positive solution for and no positive solution for . Actually, Adimurthi et al. [2], Damascelli et al. [3], Ouyang and Shi [4], and Tang [5] proved that there exists such that () in the unit ball has exactly two positive solutions for and 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 Ambrosetti et al. [6], GarcΓ­a Azorero et al. [7], Brown and Wu [8], Brown and Zhang [9], Cao and Zhong [10], de Figueiredo et al. [11], and their references. However, little has been done for this type of problem in . We are only aware of the works [12–16] 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 [17]. Wu in [17] has 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.

The main aim of this paper is to study () in involving concave-convex nonlinearities and sign-changing weight functions. We will discuss the Nehari manifold and examine carefully connection between the Nehari manifold and the fibrering maps; then using arguments similar to those used in [18], we will prove the existence of two positive solutions by using Ekeland’s variational principle [19].

Set where , , and is the best Sobolev constant for the imbedding of into . Now, we state the first main result about the existence of positive solution of () in .

Theorem 1.1. Assume that (A1) and (B1) hold. If , then () admits at least one positive solution in .

Associated with (), we consider the energy functional in : where . By [20, Proposition B.10], . It is well known that the solutions of () are the critical points of the energy functional in .

Under assumptions (A1), (B1), and , () can be regarded as a perturbation problem of the following semilinear elliptic equation: where and for all . We denote by the best constant which is given by A typical approach for solving problem of this kind is to use the Minimax method: where, and . By the Mountain Pass Lemma due to Ambrosetti and Rabinowitz [21], we called the nonzero critical point of a ground state solution of () in if . We remark that the ground state solutions of () in can also be obtained by the Nehari minimization problem where . Note that contains every nonzero solution of () in (see Willem [22])

When is a constant function in , the existence of ground state solutions of () in has been established by Berestycki and Lions [23]. Actually, Kwong [24] proved that the positive solution of () in is unique.

When and , it is well known that the existence of ground state solutions of () has been established by the condition and the existence of ground state solutions of limit equation

In order to get the second positive solution of () in , we need some additional assumptions for and . We assume the following conditions on and : for all , and satisfies suitable conditions such that () in has a positive ground state solution , that is, ; , where is a positive ground state solution of () in .

Theorem 1.2. Assume that (A1)-(A2) and (B1)-(B2) hold. If , () admits at least two positive solutions in .

Remark 1.3. (i) In [17, Theorem ], the author has proved that if where then for sufficiently small and (), admits at least two positive solutions in In particular, satisfies the following condition:
() According to Lions' paper, if for any then there is a positive ground state solution of () in Supposing we can prove that for sufficiently small (), admits at least two positive solutions in We give an example of as follows. Let be a -function on such that and Since there is a positive number such that Let be a -function on such that and Define then by (1.13), we have that In this case, and do not satisfy the assumptions of exponential decay in [17].

Throughout this paper, (A1) and (B1) will be assumed. denotes the standard Sobolev space, whose norm is induced by the standard inner product. The dual space of will be denoted by . denotes the usual scalar product in . We denote the norm in by for . denotes as . , will denote various positive constants, the exact values of which are not important. This paper is organized as follows. In Section 2, we give some properties of Nehari manifold. In Sections 3 and 4, we complete proofs of Theorems 1.1 and 1.2.

2. Nehari Manifold

In this section, we will give some properties of Nehari manifold. As the energy functional is not bounded below on , it is useful to consider the functional on the Nehari manifold Thus, if and only if Note that contains every nonzero solution of (). Moreover, we have the following results.

Lemma 2.1. The energy functional is coercive and bounded below on .

Proof. If , then by (A1), (2.2), and HΓΆlder and Sobolev inequalities Thus, is coercive and bounded 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 DrΓ‘bek and Pohozaev in [25] and are also discussed in [9]. 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. Assume that is a local minimizer for on and . Then in .

Proof. Our proof is almost the same as that in Brown and Zhang [9, Theorem ] (or see Binding et al. [26]).

Lemma 2.3. One has the following.(i)If , then ; (ii)If , then .

Proof. The proof is immediate from (2.8) and (2.9).

Moreover, we have the following result.

Lemma 2.4. If , then , where is the same as in (1.1).

Proof. Suppose the contrary. Then there exists such that . Then for by (2.8) and Sobolev inequality, we have and so Similarly, using (2.9) and HΓΆlder and Sobolev inequalities, we have which implies that Hence, we must have which is a contradiction. This completes the proof.

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 . Suppose that . Then, by (2.16), 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 2.5. Let . For each , one has the following. (i)If , then there exists a unique such that , is increasing on and decreasing on . Moreover, (ii)If , then there exists unique such that , , is decreasing on , increasing on , and decreasing on (iii). (iv)There exists a continuous bijection between and . In particular, is a continuous function for .

Proof. Fix . (i)Suppose . Then has a unique solution such that and . Thus, by , has a unique critical point at and . Therefore, and (2.19) holds. (ii)Suppose . Since , the equation has exactly two solutions such that and . Thus, there exist exactly two multiples of lying in , that is, and . Therefore, by , has critical points at and with and . Therefore, is decreasing on , increasing on and decreasing on . This implies that (2.20) holds. (iii)For . By Lemma 2.3(ii) and, considering , we have . By (i) and (ii), there exists a unique such that , that is, . Since , we have . Therefore Conversely, if is such that , then by the uniqueness of , we get that . Thus, we have (iv)Fix arbitrary. Define by where is defined by . Since , and then by the implicit function theorem, there is a neighborhood of in and a unique continuous function such that for all , in particular . Since is arbitrary, we obtain that the function , given by , is continuous and one to one. By , where , we have that is continuous and one to one. Now if , then by (iii) we have that , where . Since is continuous on , it follows that is continuous on . This completes the proof.

3. Proof of Theorem 1.1

First, we remark that it follows from Lemma 2.4 that for all . Furthermore, by Lemma 2.5 it follows that and are nonempty, and by Lemma 2.1 we may define Then we get the following result.

Theorem 3.1. One has the following.(i)If , then one has . (ii)If , then for some . In particular, for each , one has .

Proof. (i) Let . By (2.8) and so Therefore, .
(ii) Let . By (2.8) Moreover, by (B1) and Sobolev inequality This implies that By (2.4) and (3.7), we have Thus, if , then for some positive constant . This completes the proof.

Remark 3.2. (i) If , then by (2.8) and HΓΆlder and Sobolev inequalities, for each we have and so
() If , then by Lemma 2.5(i), (ii) and Theorem 3.1(ii), for each we have

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

Definition 3.3. For , a sequence is a sequence in for if and strongly in as
   is a value in for if there exists a sequence in for .
   satisfies the -condition in if any sequence in for contains a convergent subsequence.

Now, we use the Ekeland variational principle [19] to get the following results.

Proposition 3.4. If , then there exists a sequence in for .
If , then there exists a sequence in for .

Proof. The proof is almost the same as that in [27, Proposition ].

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

Theorem 3.5. Assume that (A1) and (B1) hold. If , then has a minimizer in , and it satisfies the following:(i); (ii) is a positive solution of () in ; (iii) as .

Proof. By Proposition 3.4(i), there is a minimizing sequence for on such that Since is coercive on (see Lemma 2.1), we get that is bounded in . Going if necessary to a subsequence, we can assume that there exists such that By (A1), Egorov theorem, and HΓΆlder inequality, we have First, we claim that is a nonzero solution of (). By (3.13) and (3.14), it is easy to see that is a solution of (). From and (2.3), we deduce that Let in (3.16); by (3.13), (3.15), and , we get Thus, is a nonzero solution of (). Now we prove that strongly in and . By (3.16), if , then In order to prove that , it suffices to recall that , by (3.18) and applying Fatou’s lemma to get This implies that and Let , then Brezis-Lieb lemma [28] implies that Therefore, strongly in . Moreover, we have . On the contrary, if