Research

# Existence and multiplicity of positive bound states for Schrödinger equations

Sun Sheng, Fanglei Wang* and Tianqing An

Author Affiliations

College of Science, Hohai University, Nanjing, 210098, P.R. China

For all author emails, please log on.

Boundary Value Problems 2013, 2013:271  doi:10.1186/1687-2770-2013-271

 Received: 5 September 2013 Accepted: 19 November 2013 Published: 12 December 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we study the existence and multiplicity of positive bound states of non-autonomous 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 theorems

### 1 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 [1-5]. 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 Leray-Schauder, 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 [4-6] 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 non-autonomous Schrödinger equations

(1)

where

The methods used here are the Krasnoselskii fixed point theorem and the Leggett-Williams 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:

(i) for every;

(ii) for every;

(iii) Given a nonempty compact subset, we have

where.

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

(i) , and, ; or

(ii) , and, .

ThenThas a fixed point in.

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 all 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

(i) andfor;

(ii) for;

(iii) forwith.

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

whereverifies

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,

Therefore, it is clear that .

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

Set . Then, for any , we have

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

Set . Then, for any , we have

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

(H5) , .

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

(H7) , for some.

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:

(H8) forand;

(H9) there existssuch that

where;

(H10) forand.

Then (1) has at least three positive solutions.

Proof For , define

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

So .

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 Leggett-Williams 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.

### References

1. Ambrosetti, A, Colorado, E: Standing waves of some coupled nonlinear Schrödinger equations. J. Lond. Math. Soc.. 75, 67–82 (2007). Publisher Full Text

2. Ambrosetti, A: Remarks on some systems of nonlinear Schrödinger equations. J. Fixed Point Theory Appl.. 4, 35–46 (2008). Publisher Full Text

3. Ambrosetti, A, Cerami, G, Ruiz, D: Solitons of linearly coupled systems of semilinear non-autonomous equations on . J. Funct. Anal.. 254, 2816–2845 (2008). Publisher Full Text

4. Torres, PJ: Guided waves in a multi-layered optical structure. Nonlinearity. 19, 2103–2113 (2006). Publisher Full Text

5. Belmonte-Beitia, J, Pérez-García, VM, Torres, PJ: Solitary waves for linearly coupled nonlinear Schrödinger equations with inhomogeneous coefficients. J. Nonlinear Sci.. 19, 437–451 (2009). Publisher Full Text

6. Chu, J: Positive bound states of systems of nonlinear Schrödinger equations. Nonlinear Anal.. 72, 1983–1992 (2010). Publisher Full Text

7. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones, Academic Press, New York (1988)

8. Zima, M: On positive solutions of boundary value problems on the half-line. J. Math. Anal. Appl.. 259, 127–136 (2001). Publisher Full Text