Abstract
Keywords:
semilinear elliptic systems; subcritical exponents; Nehari manifold1 Introduction
For , , and , we consider the semilinear elliptic systems
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
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 [25] 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 PalaisSmale 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
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
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
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.
2 Preliminaries
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 .
then . Fix some , there exists such that and . Let . □
Define
Lemma 2.4For each, there exists a unique positive numbersuch thatand.
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
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 .
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 . □
3 (PS)_{γ}condition in H for
First of all, we define the PalaisSmale (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 CaoZhou [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 BrézisLieb 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
This implies . By (3.6) and (3.7), we obtain that
which is a contradiction. Hence, , that is, strongly in H. □
4 Existence of k solutions
Let be the unique, radially symmetric and positive ground state solution of equation (E0) in . Recall the facts (or see BahriLi [10], BahriLions [11], GidasNiNirenberg [12] and Kwong [13]):
(ii) for any , there exist positive numbers , and such that for all
and
By LienTzengWang [14], then
First of all, we want to prove that
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
Since
then
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. □
where and , define and . Suppose for some . Let be given by
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
□
We need the following lemmas to prove that for sufficiently small ε, λ, μ.
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 concentrationcompactness 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 ,
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
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 asequenceinHfor.
Proof See CaoZhou [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. □
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The author was grateful for the referee’s helpful suggestions and comments.
References

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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