#### Abstract

We study the effect of the coefficient of the critical nonlinearity on the number of positive solutions for a --Laplacian equation. Under suitable assumptions for and , we should prove that for sufficiently small , there exist at least positive solutions of the following --Laplacian equation, , where is a bounded smooth domain, , , is the critical Sobolev exponent, and is the -Laplacian of .

#### 1. Introduction

This paper is concerned with the multiplicity of positive solutions to the following --Laplacian equation with critical nonlinearities: where is a bounded smooth domain with smooth boundary , , , and is the -Laplacian of , and assume that; and are positive continuous functions in ;There exist points in such that are strict local maxima satisfying â€‰and for some , as uniformly in .

Problem comes, for example, from a general reaction-diffusion system where . This system has a wide range of applications in physics and related science such as biophysics, plasma physics, and chemical reaction design. In such applications, the function describes a concentration, the first term on the right-hand side of (2) corresponds to the diffusion with a diffusion coefficient , whereas the second one is the reaction and relates to sources and loss processes. Typically, in chemical and biological applications, the reaction term has a polynomial form with respect to the concentration .

The stationary solution of (2) was studied by many authors; that is, many works are considered the solutions of the following problem: See [1â€“5] for different . In the present paper we are concerned with problem in a bounded domain with in (3). Recently, in [6], the authors obtain the existence of positive solutions of problem for and when condition holds, where denotes the Lusternik-Schnirelmann category of in itself.

Specially, if , can be reduced to the following elliptic problems:

After the well-known results of BrÃ©zis and Nirenberg [7], who studied (4) in the case of and , a lot of problems involving the critical growth in bounded and unbounded domains have been considered; see, for example, [8â€“10] and reference therein. In particular, the first multiplicity result for (4) has been achieved by Rey in [11] in the semilinear case. Precisely Rey proved that if , , and , for small enough, problem (4) has at least solutions. Furthermore, Alves and Ding in [12] obtained the existence of positive solutions to problem (4) with , , and . Finally, we mention that [13] studied (4) when and are sign-changing and verified the existence of two positive solutions for for some positive constant .

The main purpose of this paper is to analyze the effect of the coefficient of the critical nonlinearity to prove the multiplicity of positive solutions of problem for small . By the similar argument in [14], we can construct the compact Palais-Smale sequences that are suitably localized in correspondence of maximum points of . Under some assumptions , we could show that there are at least positive solutions of problem for sufficiently small .

This paper is organized as follows. First of all, we study the argument of the Nehari manifold . Next, we prove the existence of a positive solution . Finally, we show that the condition affects the number of positive solution of ; that is, there are at least critical points of such that ((PS)-value) for .

The main results of this paper are given as follows.

Theorem 1. *Suppose that (H1)â€“(H3) hold; then problem has a positive solution in for all .*

Theorem 2. *Suppose that (H1)â€“(H3) hold; then there exists a such that for any , problem admits at least positive solutions in .*

#### 2. Preliminaries

In what follows, we denote by the norms on and , respectively; that is, We denote the dual space of by . Set also equipped with the norm We will denote by the best Sobolev constant as follows: It is well known that is independent of and is never achieved except when (see [15]). Throughout this paper, we denote the Lebesgue measure of by and denote a ball centered at with radius by and also denote positive constants (possibly different) by , . denotes , denotes as , and denotes as .

Associated with , we consider the energy functional in , for each , It is well known that is of in and the solutions of are the critical points of the energy functional (see [16]).

We define the Nehari manifold where The Nehari manifold contains all nontrivial solutions of .

Note that is not bounded from below in . From the following lemma, we have that is bounded from below on the Nehari manifold .

Lemma 3. *Suppose that and (H2) hold. Then for any , one has that is bounded from below on . Moreover, for all .*

*Proof. *For , (10) leads to

Define

Now we show that possesses the mountain-pass (MP, in short) geometry.

Lemma 4. *Suppose and (H2) holds. Then for any , one has that*(i)*there exist positive numbers and such that for ;*(ii)*there exists such that and .*

*Proof. * By (8), the HÃ¶lder inequality, and the Sobolev embedding theorem, we have that
Hence, there exist positive and such that for .

For any , from
we have . For fixed some , there exist such that and . Let .

Define Then for ,

Lemma 5. *Suppose that and (H2) holds. If satisfies
**
then is a solution of .*

*Proof. *By (17), for . Since , by the Lagrange multiplier theorem, there is such that in . This implies that
It then follows that and in . Thus, is a nontrivial solution of and .

Lemma 6. *Suppose that and (H2) holds. For each , there exists a unique positive number such that and for any .*

*Proof. *For fixed , consider
Since , , by Lemma 4, then it is easy to see that there exists a unique positive number such that is achieved at . This means that ; that is, .

We will denote by the MP level:

Then we have the following result.

Lemma 7. *Suppose that and (H2) holds, then for any .*

*Proof. *By Lemma 6, we have
On the other hand, for , by Lemma 6, we have and . Hence,
Now the desired result follows from (22) and (23).

*Remark 8. *By Lemma 7 and the definition, it is apparent that if ; that is, is nonincreasing in . Moreover, by Lemma 4(i), for any , there exists a , related to the MP geometry, such that
Here is the MP level associated to the functional

#### 3. PS-Condition in for

First, we define the Palais-Smale (denote by (PS)) sequence, (PS)-value, and (PS)-conditions in for .

*Definition 9. *(i) For , a sequence is a -sequence in for if and strongly in as .

(ii) is a (PS)-value in for if there exists a -sequence in forâ€‰â€‰.

(iii) satisfies the -condition in if every -sequence in for contains a convergent subsequence.

Applying Ekelandâ€™s variational principle and using the same argument as in Cao and Zhou [17] or Tarantello [18], we have the following lemma.

Lemma 10. *Suppose that and (H2) holds. Then for any , there exists a -sequence in for .*

To prove the existence of positive solutions, we claim that satisfies the -condition in for .

Lemma 11. *Suppose that , , and (H2) holds. Then for any , satisfies the (PS) _{c}-condition in for all .*

*Proof. *Let be a -sequence for which satisfies
Then
where , , as . It follows that is bounded in . Thus, there exist a subsequence still denoted by and such that
Furthermore, we have that in . By being continuous on , we get
Let . Then by being positive continuous on and BrÃ©zis-Lieb lemma (see [19]), we obtain
From (26)â€“(30), we can deduce that
Without loss of generality, we may assume that
So (32) and imply that
By the Sobolev inequality and (33) and (34), we have and
If , then (35) implies that , combined with (31), (33)â€“(35) and Lemma 3, , as ; we get
which is a contradiction. So, we have ; satisfies the -condition in for all .

#### 4. Existence of Positive Solutions

In this section, we first give some preliminary notations and useful lemmas.

Choose small enough such that and for ,â€‰â€‰.

Define Then we have the following separation result.

Lemma 12. *If and for , then .*

*Proof. *For any satisfying , we get
which implies that
Hence, from (39), we obtain
which is a contradiction.

For , we set and define Now let us assume that hold. From conditions and , we can choose a small enough and there exist some positive constants such that for , we have for some . For and , we define where is a function such that and on . Then we obtain the following estimates (see [20]): From (43)â€“(46) [13, Lemma 4.2] and conditions -, we can deduce the following estimates: where , , and are positive constants independent of , and â€‰â€‰is the best Sobolev constant given in (8).

Next, we will investigate the effect of the coefficient to find some Palais-Smale sequences which are used to prove Theorem 2.

Lemma 13. *If (H1)â€“(H3) hold, then for any and any , there exists a such that for one has
**
In particular, for all .*

*Proof. *By Lemma 6, there exists a such that . Furthermore,
where and . Hence, there exists an small enough such that for any , we have
which implies for any , and then
Set
Since , , then there exists a such that hold, and then satisfies
then we have
From (47) and (48), fixing any small enough, there exists such that
Also, from (55), we obtain
From (47)â€“(49) and (58), there exist and such that
Let ; then
where and are independent of . From [13, Lemma 4.2] and conditions , we also have
By (43), (46)â€“(49), (60), and (61), for , we obtain
where are positive constants independent of . Since , we obtain that
then there exists an such that uniformly in for all . Moreover, from (53), we have for all and . This completes the proof.

*Proof of Theorem 1. *From Lemmas 5, 10, 11, and 13, we get for all that there exists a such that and . Set . Replace the terms and of the functional by and , respectively. It then follows that is a nonnegative solution of . Applying the maximum principle, admits at least one positive solution in .

By studying the argument as in [21, Theorem III 3.1] and [22], we obtain the following lemma.

Lemma 14. *Let be a nonnegative function sequence with and . Then there exists a sequence such that
**
contains a convergent subsequence denoted again by such that
**
where in . Moreover, we have , , and as .*

Lemma 15. *Suppose that (H2) and (H3) hold. Then for any , there exists such that
*

*Proof. *Fix . Assume the contrary; that is, there then exists a sequence with as such that . Consequently, there exists a sequence such that, as ,
and by Remark 8, we have that there exists a such that