Multiple positive solutions for semilinear elliptic systems involving subcritical nonlinearities in R N

In this paper, we investigate the effect of the coefficient of the subcritical nonlinearity. Under some assumptions, for sufficiently small , there are at least k (≥1) positive solutions of the semilinear elliptic systems

where , , for .

MSC: 35J20, 35J25, 35J65.

For , , and , we consider the semilinear elliptic systems

where .

Let f, g and h satisfy the following conditions:

(A1) f is a positive continuous function in and .

(A2) there exist k points in such that

and .

(A3) where , and .

In [1], if Ω is a smooth and bounded domain in (), they considered the following system:

and proved the existence of a least energy solution in Ω for sufficiently small and . Lin and Wei also showed that this system has a least energy solution in for and . In this paper, we study the effect of of (). Recently, many authors [2-5] considered the elliptic systems with subcritical or critical exponents, and they proved the existence of a least energy positive solution or the existence of at least two positive solutions for these problems. In this paper, we construct the k compact Palais-Smale sequences which are suitably localized in correspondence of k maximum points of f. Then we could show that under some assumptions (A1)-(A3), for sufficiently small , there are at least k (≥1) positive solutions of the elliptic system (). By the change of variables

System () is transformed to

Let be the space with the standard norm

Associated with the problem (), we consider the -functional , for ,

Actually, the weak solution of () is the critical point of the functional , that is, satisfies

for any .

We consider the Nehari manifold

where

The Nehari manifold contains all nontrivial weak solutions of ().

Let

then by [[2], Theorem 5], we have

where and is the best Sobolev constant defined by

For the semilinear elliptic systems ()

we define the energy functional , and

If (=1), then we define and

where .

It is well known that this problem

has the unique, radially symmetric and positive ground state solution . Define and , where

Moreover, we have that

This paper is organized as follows. First of all, we study the argument of the Nehari manifold . Next, we prove that the existence of a positive solution of (). Finally, in Section 4, we show that the condition (A2) affects the number of positive solutions of (); that is, there are at least k critical points of such that ((PS)-value) for .

Theorem 1.1 () has at least one positive solution, that is, () admits at least one positive solution.

Theorem 1.2There exist two positive numbersandsuch that () has at leastkpositive solutions for anyand, that is, () admits at leastkpositive solutions.

By studying the argument of Han [[7], Lemma 2.1], we obtain the following lemma.

Lemma 2.1Let (possibly unbounded) be a smooth domain. If, weakly in, and, almost everywhere in Ω, then

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

Lemma 2.2The energy functionalis bounded from below on.

Proof For , by (1.1), we obtain that

where . Hence, we have that is bounded from below on . □

We define

Lemma 2.3 (i) There exist positive numbersσandsuch thatfor;

(ii) There existssuch thatand.

Proof (i) By (1.2), the Hölder inequality (, ) and the Sobolev embedding theorem, we have that

where and . Hence, there exist positive σ and such that for .

(ii) For any , since

then . Fix some , there exists such that and . Let . □

Define

Then for , we obtain that

Lemma 2.4For each, there exists a unique positive numbersuch thatand.

Proof Fixed , we consider

Since , , by Lemma 2.3(i), then is achieved at some . Moreover, we have that , that is, . Next, we claim that is a unique positive number such that . Consider

then . Since ,

there exists a unique positive number such that . It follows that . Hence, . □

Remark 2.5 By Lemma 2.3(i) and Lemma 2.4, then for some constant .

Lemma 2.6Letsatisfy

thenis a solution of ().

Proof By (2.2), for . Since , by the Lagrange multiplier theorem, there is such that in . Then we have

It follows that and in . Therefore, is a nontrivial solution of () and . □

First of all, we define the Palais-Smale (denoted by (PS)) sequence and (PS)-condition in H for some functional J.

Definition 3.1 (i) For , a sequence is a (PS)_γ-sequence in H for J if and strongly in as , where is the dual space of H;

(ii) J satisfies the (PS)_γ-condition in H if every (PS)_γ-sequence in H for J contains a convergent subsequence.

Applying Ekeland’s variational principle and using the same argument as in Cao-Zhou [8] or Tarantello [9], we have the following lemma.

Lemma 3.2 (i) There exists a (PS)-sequenceinfor.

In order to prove the existence of positive solutions, we want to prove that satisfies the (PS)_γ-condition in H for .

Lemma 3.3satisfies the (PS)_γ-condition inHfor.

Proof Let be a (PS)_γ-sequence in H for such that and in . Then

where , as . It follows that is bounded in H. Hence, there exist a subsequence and such that

Moreover, we have that in . We use the Brézis-Lieb lemma to obtain (3.1) and (3.2) as follows:

Next, we claim that

and

Since , where , then for any , there exists such that . By the Hölder inequality and the Sobolev embedding theorem, we get

Similarly, as . By (A1) and , strongly in , we have that

Let . By (3.1)-(3.4) and Lemma 2.1, we deduce that

and

We may assume that

Recall that

If , by (3.5), then

This implies . By (3.6) and (3.7), we obtain that

which is a contradiction. Hence, , that is, strongly in H. □

Let be the unique, radially symmetric and positive ground state solution of equation (E0) in . Recall the facts (or see Bahri-Li [10], Bahri-Lions [11], Gidas-Ni-Nirenberg [12] and Kwong [13]):

(i) for some and ;

(ii) for any , there exist positive numbers , and such that for all

and

By Lien-Tzeng-Wang [14], then

For , we define

Clearly, .

First of all, we want to prove that

Lemma 4.1Forand, then

Moreover, we have that

Proof Part I: Since is continuous in H, , and is uniformly bounded in H for any and , then there exists such that for and any ,

From (A1), we have that . Then

It follows that there exists such that for and any ,

From now on, we only need to show that

Since

and by (4.1), then

For , by (4.2), we have that

Since

then

that is, for and ,

Part II: By Lemma 2.4, there is a number such that , where . Hence, from the result of Part I, we have that for and ,

 □

Proof of Theorem 1.1 By Lemma 3.2, there exists a (PS)-sequence in for . Since for and , by Lemma 3.3, there exist a subsequence and such that strongly in H. It is easy to check that is a nontrivial solution of () and . Since and , by Lemma 2.6, we may assume that , . Applying the maximum principle, and in Ω. □

Choosing such that

where and , define and . Suppose for some . Let be given by

where , for and for .

For each , we define

By Lemma 2.4, there exists such that for each . Then we have the following result.

Lemma 4.2There existssuch that if, thenfor each.

Proof Since

there exists such that

 □

We need the following lemmas to prove that for sufficiently small ε, λ, μ.

Lemma 4.3.

Proof From Part I of Lemma 4.1, we obtain uniformly in i. Similarly to Lemma 2.4, there is a sequence such that and

Let be a minimizing sequence of for . It follows that and

We may assume that and as , where . By the definition of , then . We can deduce that , that is, . □

Lemma 4.4There exists a numbersuch that ifand, thenfor any.

Proof On the contrary, there exist the sequences and such that , as and for all . It is easy to check that is bounded in H. Suppose that as . Since

then

which is a contradiction. Thus, as . Similarly to the concentration-compactness principle (see Lions [15,16] or Wang [[6], Lemma 2.16]), then there exist a constant and a sequence such that

where and for some . Let . Then there are a subsequence and such that and weakly in . Using the similar computation of Lemma 2.4, there is a sequence such that and

We deduce that a subsequence satisfies . Then there are a subsequence and such that and weakly in . By (4.3), then and . Applying Ekeland’s variational principle, there exists a (PS)-sequence for and . Similarly to the proof of Lemma 3.3, there exist a subsequence and such that , strongly in and . Now, we want to show that there exists a subsequence such that .

(i) Claim that the sequence is bounded in . On the contrary, assume that , then

which is a contradiction.

(ii) Claim that On the contrary, assume that , that is, . Then use the argument of (i) to obtain that

which is a contradiction.

Since and , strongly in , we have that

which is a contradiction.

Hence, there exists such that if and , then for any . □

Lemma 4.5Ifand, then there exists a numbersuch thatfor anyand.

Proof Using the similar computation in Lemma 2.4, we obtain that there is the unique positive number

such that . We want to show that there exists such that if , then for some constant (independent of u and v). First, for ,

Since , then

and

Moreover, we have that

where . It follows that there exists such that for

Hence, by (4.4), (4.5) and (4.6), for some constant (independent of u and v) for . Now, we obtain that

From the above inequality, we deduce that for any and ,

Hence, there exists such that for ,

By Lemma 4.4, we obtain

or for any and . □

Since , then by Lemma 4.3,

By Lemmas 4.1, 4.2 and (4.7), for any () and ,

Applying above Lemma 4.5, we get that

For each , by (4.8) and (4.9), we obtain that

It follows that

Then applying Ekeland’s variational principle and using the standard computation, we have the following lemma.

Lemma 4.6For each, there is a-sequenceinHfor.

Proof See Cao-Zhou [8]. □

Proof of Theorem 1.2 For any and , by Lemma 4.6, there is a -sequence for where . By (4.8), we obtain that

Since satisfies the (PS)_γ-condition for , then has at least k critical points in for any and . Set and . Replace the terms and of the functional by and , respectively. It follows that () has k nonnegative solutions. Applying the maximum principle, () admits at least k positive solutions. □

The author declares that he has no competing interests.

The author was grateful for the referee’s helpful suggestions and comments.

1. Lin, TC, Wei, J: Spikes in two coupled nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 22, 403–439 (2005). Publisher Full Text

2. Alves, CO, de Morais Filho, DC, Souto, MAS: On systems of elliptic equations involving subcritical or critical Sobolev exponents. Nonlinear Anal.. 42, 771–787 (2000). Publisher Full Text

3. Han, P: Multiple positive solutions of nonhomogeneous elliptic system involving critical Sobolev exponents. Nonlinear Anal.. 64, 869–886 (2006). Publisher Full Text

4. Hsu, TS: Multiple positive solutions for a critical quasilinear elliptic system with concave-convex nonlinearities. Nonlinear Anal.. 71, 2688–2698 (2009). Publisher Full Text

5. Wu, TF: The Nehari manifold for a semilinear elliptic system involving sign-changing weight function. Nonlinear Anal.. 68, 1733–1745 (2008). Publisher Full Text

6. Wang, HC: Palais-Smale approaches to semilinear elliptic equations in unbounded domains. Electron. J. Differ. Equ. Monograph 06 (2004)

7. Han, P: The effect of the domain topology on the number of positive solutions of elliptic systems involving critical Sobolev exponents. Houst. J. Math.. 32, 1241–1257 (2006)

8. Cao, DM, Zhou, HS: Multiple positive solutions of nonhomogeneous semilinear elliptic equations in . Proc. R. Soc. Edinb., Sect. A, Math.. 126, 443–463 (1996). Publisher Full Text

9. Tarantello, G: On nonhomogeneous elliptic involving critical Sobolev exponent. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 9, 281–304 (1992)

10. Bahri, A, Li, YY: On a min-max procedure for the existence of a positive solution for certain scalar field equations in . Rev. Mat. Iberoam.. 6, 1–15 (1990)

11. Bahri, A, Lions, PL: On the existence of a positive solution of semilinear elliptic equations in unbounded domains. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 14, 365–413 (1997). Publisher Full Text

12. Gidas, B, Ni, WM, Nirenberg, L: Symmetry and related properties via the maximum principle. Commun. Math. Phys.. 68, 209–243 (1979). Publisher Full Text

13. Kwong, MK: Uniqueness of positive solutions of in . Arch. Ration. Mech. Anal.. 105, 234–266 (1989)

14. Lien, WC, Tzeng, SY, Wang, HC: Existence of solutions of semilinear elliptic problems on unbounded domains. Differ. Integral Equ.. 6, 1281–1298 (1993)

15. Lions, PL: The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 1, 109–145 (1984)

16. Lions, PL: The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 1, 223–283 (1984)