Multiplicity of Positive Solutions for a --Laplacian Type Equation with Critical Nonlinearities
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 .
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 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 [1–5] for different . In the present paper we are concerned with problem in a bounded domain with in (3). Recently, in , 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 Brézis and Nirenberg , 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, [8–10] and reference therein. In particular, the first multiplicity result for (4) has been achieved by Rey in  in the semilinear case. Precisely Rey proved that if , , and , for small enough, problem (4) has at least solutions. Furthermore, Alves and Ding in  obtained the existence of positive solutions to problem (4) with , , and . Finally, we mention that  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 , 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)–(H3) hold; then problem has a positive solution in for all .
Theorem 2. Suppose that (H1)–(H3) hold; then there exists a such that for any , problem admits at least positive solutions in .
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 ). 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 ).
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
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 Hölder 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 for .
(iii) satisfies the -condition in if every -sequence in for contains a convergent subsequence.
Applying Ekeland’s variational principle and using the same argument as in Cao and Zhou  or Tarantello , 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 Brézis-Lieb lemma (see ), we obtain From (26)–(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)–(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 , .
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 ): From (43)–(46) [13, Lemma 4.2] and conditions -, we can deduce the following estimates: where , , and are positive constants independent of , and 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)–(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)–(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)–(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 , 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