Abstract
In this paper, we study the existence and multiplicity of positive bound states of nonautonomous systems of nonlinear Schrödinger equations. The proof is based on the fixed point theorems in a cone.
MSC: 34B15, 35J20.
Keywords:
nonlinear Schrödinger systems; positive solutions; fixed point theorems1 Introduction
Because of the important background in nonlinear optics and other fields, many authors pay more attention to the study of different types of vector nonlinear Schrödinger equations, we refer the readers to [15]. Most of these results have been proven using critical point theory, variational approaches or a fixed point theorem. More recently, Chu has applied another topological approach, a nonlinear alternative principle of LeraySchauder, to establish some new existence results for the following Schrödinger equations
where is nonnegative almost everywhere, . The author considered two different cases. One is the singular case, that is, and
The other is the regular case, that is, . However, the references [46] are not concerned with multiplicity of positive solutions for the scalar Schrödinger equation or system.
Motivated by the study of solitary wave solutions, in this paper we mainly aim to study the existence and multiplicity of positive bound states of the more general system of nonautonomous Schrödinger equations
where
The methods used here are the Krasnoselskii fixed point theorem and the LeggettWilliams fixed point theorem together with a compactness criterion due to Zima.
We organize the paper as follows. In Section 2, we give some preliminaries; in Section 3, we discuss the existence and multiplicity of positive solutions for (1).
2 Preliminaries
For convenience, we assume that the following conditions hold throughout this paper.
(H1) , and satisfies the following property
(H2) The support of denoted by is a nonempty compact set, and
(H3) is continuous, and the following notations are introduced:
Since (H1) holds, then for the homogeneous problem
the associated Green’s function is expressed by
where , are solutions such , . Moreover, , can be chosen as positive increasing and positive decreasing functions, respectively. Note that , intersect at a unique point . Therefore, we can define a function by
where denotes the space of bounded continuous functions.
Lemma 2.1[4]
For each, Green’s functionsatisfies the following properties:
(iii) Given a nonempty compact subset, we have
Lemma 2.2[4]
Assume that (H1) holds and. Then the unique solution of
belongs to, and the solution can be expressed as
The proof of our main results is based on the following fixed points, which can be found in [7].
Lemma 2.3LetEbe a Banach space, and letbe a cone inE. Assume that, are open subsets ofEwith, , and letbe a completely continuous operator such that either
Let E be a real Banach space and P be a cone in E. A map α is said to be a nonnegative continuous concave functional on P if
is continuous and
For numbers a, b such that , letting α be a nonnegative continuous concave functional on P, we define the following convex sets:
and
Lemma 2.4Letbe completely continuous andαbe a nonnegative continuous concave functional onPsuch thatfor all. Suppose that there existsuch that
ThenThas at least three fixed points, , satisfying
In addition, the following compactness criterion proved by Zima in [8] is also used in our proof.
Lemma 2.5Let. Let us assume that the functionsare equicontinuous in each compact interval ofRand that for all, we have
Then Ω is relatively compact.
3 Main results
From now on, we assume that is a nonempty compact set. Let E denote the Banach space with the norm , for . Define a cone as
where , and the constants , , are defined by property (iii) of Lemma 2.1. Since M is compact, then , . Moreover, from (iii) of Lemma 2.1 it follows that for .
Let be a map with components defined by
A fixed point of T is a solution of (1) which belongs to .
Lemma 3.1Assume that (H1)(H3) hold. Then, andis completely continuous.
Proof The continuity is trivial. Since M is compact, there exists a point where is attained. Then, for any , we have
namely,
Finally, we prove that each component of T is compact. Let be a bounded set, then there exists a constant which is uniformly bounded for its element. Since the derivative is bounded in compacts, the functions of are equicontinuous on each compact interval. On the other hand, for any ,
□
Theorem 3.2Assume that (H1)(H3) hold. In addition, () satisfies
(a) If, for some, then (1) has at least one positive solution.
(b) If, for some, then (1) has at least one positive solution.
Proof (a) On the one hand, since , then there exists such that
where is sufficiently small such that
Furthermore, for any , we have
On the other hand, since for some , then there exists such that
where is sufficiently large such that
Let and set . Then, for any , , and we have
Furthermore, we have
Now by Lemma 2.3, T has a fixed point , namely, (1) has a positive solution.
(b) On the one hand, since for some , there exists such that
where is sufficiently large such that
Furthermore, we have
On the other hand, since , then there exists such that
where is sufficiently small such that
Let and set . Then, for any , , and we have
Furthermore, for any , we have
Now, by Lemma 2.3, T has a fixed point , namely, (1) has a positive solution. □
Corollary 3.3Assume that (H1)(H3) hold. () satisfies
In addition, the following conditions hold.
(H4) If there exist constantssuch that for some,
whereρsatisfies
Then (1) has at least two positive solutions.
Corollary 3.4Assume that (H1)(H3) hold. () satisfies
In addition, the following conditions hold.
(H6) If there exist constantssuch that
whereϑsatisfies
Then (1) has at least two positive solutions.
Theorem 3.5Assume that (H1)(H3) hold and () satisfies
In addition, there exist numbersa, canddwithsuch that the following conditions are satisfied:
Then (1) has at least three positive solutions.
then it is easy to know that α is a nonnegative continuous concave functional on K with for .
Set . First, we show that with . For any , we have
In a similar way, we also can prove that . Then (ii) of Lemma 2.4 holds.
Next, we shall show that (i) of Lemma 2.4 is satisfied. It is clearly seen that . Then, for any and , it is easy to obtain that
Then, by (H9), we can have
Finally, we verify that (iii) of Lemma 2.4 is satisfied. Suppose that with , then we can have
From the above, the hypotheses of LeggettWilliams theorem are satisfied. Hence (1) has at least three positive solutions , and such that , , and with
□
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
SS drafted the manuscript. FW and TA gave some suggestions to improve the manuscript. All authors typed, read and approved the final manuscript.
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