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Existence and multiplicity of positive bound states for Schrödinger equations

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The Erratum to this article has been published in Boundary Value Problems 2016 2016:199

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.

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 [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 Leray-Schauder, to establish some new existence results for the following Schrödinger equations

u ¨ (x)+a(x)u(x)=b(x)f ( u ( x ) ) +e(x),

where a L (R, R N ) is nonnegative almost everywhere, e L 1 (R, R N ). The author considered two different cases. One is the singular case, that is, fC( R N {0}, R N ) and

lim u 0 f i (u)=+,i=1,2,,n.

The other is the regular case, that is, fC( R N , R N ). 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 non-autonomous Schrödinger equations

{ u ¨ ( x ) + A ( x ) u ( x ) = C ( x ) F ( u ( x ) ) , lim | x | u ( x ) = lim | x | u ˙ ( x ) = 0 , + u 2 ( x ) d x + + u ˙ 2 ( x ) d x < + ,
(1)

where

u ( x ) = ( u 1 ( x ) u 2 ( x ) u n ( x ) ) , A ( x ) = ( a 11 ( x ) a 12 ( x ) a 1 n ( x ) a 21 ( x ) a 22 ( x ) a 2 n ( x ) a n 1 ( x ) a n 2 ( x ) a n n ( x ) ) , C ( x ) = ( c 1 ( x ) 0 0 0 c 2 ( x ) 0 0 0 c n ( x ) ) , F ( u ( x ) ) = ( f 1 ( u 1 ( x ) , u 2 ( x ) , , u n ( x ) ) f 2 ( u 1 ( x ) , u 2 ( x ) , , u n ( x ) ) f n ( u 1 ( x ) , u 2 ( x ) , , u n ( x ) ) ) .

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) a i j L (R,R), and a i j (x) satisfies the following property

a i j (x)= { a i i ( x ) 0 , and  inf a i i ( x ) > 0  if  i = j , a i j ( x ) 0 , and  Supp ( a i j )  is a nonempty compact set if  i j .

(H2) The support of c i (x)>0 denoted by Supp( c i ) is a nonempty compact set, and

0< M G i (s,s) c i (s)ds<+.

(H3) f i (u(x))C( R N , R + ) is continuous, and the following notations are introduced:

f i , 0 = lim u 0 f i ( u ) i = 1 n u i , f i , = lim u f i ( u ) i = 1 n u i .

Since (H1) holds, then for the homogeneous problem

{ ϕ ¨ ( x ) + a i i ( x ) ϕ ( x ) = 0 , ϕ ( ) = 0 , ϕ ( + ) = 0 ,

the associated Green’s function is expressed by

G i (x,s)= { ϕ i 1 ( x ) ϕ i 2 ( s ) , < x s < + , ϕ i 1 ( s ) ϕ i 2 ( x ) , < s x < + ,

where ϕ i 1 , ϕ i 2 are solutions such ϕ i 1 ()=0, ϕ i 2 (+)=0. Moreover, ϕ i 1 , ϕ i 2 can be chosen as positive increasing and positive decreasing functions, respectively. Note that ϕ i 1 , ϕ i 2 intersect at a unique point x 0 . Therefore, we can define a function p i (x)BC(R) by

p i (x)= { 1 ϕ i 2 ( x ) , x x 0 , 1 ϕ i 1 ( x ) , x > x 0 ,

where BC(R) denotes the space of bounded continuous functions.

Lemma 2.1 [4]

For each i=1,2,,n, Green’s function G i (x,s) satisfies the following properties:

  1. (i)

    G i (x,s)>0 for every (x,s)R×R;

  2. (ii)

    G i (x,s) G i (s,s) for every (x,s)R×R;

  3. (iii)

    Given a nonempty compact subset PR, we have

    G i (x,s) m i (P) p i (s) G i (s,s)for all (x,s)P×R,

where m i (P)=min{ ϕ i 1 (infP), ϕ i 2 (infP)}.

Lemma 2.2 [4]

Assume that (H1) holds and e(x) L 1 (R). Then the unique solution of

{ u i ¨ ( x ) + a i i ( x ) u i ( x ) = e ( x ) , u i ( ) = 0 , u i ( + ) = 0

belongs to H 1 (R), and the solution can be expressed as

u i (x)= R G i (x,s)e(s)ds.

The proof of our main results is based on the following fixed points, which can be found in [7].

Lemma 2.3 Let E be a Banach space, and let KE be a cone in E. Assume that Ω 1 , Ω 2 are open subsets of E with 0 Ω 1 , Ω ¯ 1 Ω 2 , and let T:K( Ω ¯ 2 Ω 1 )K be a completely continuous operator such that either

  1. (i)

    Tuu, uK Ω 1 and Tuu, uK Ω 2 ; or

  2. (ii)

    Tuu, uK Ω 1 and Tuu, uK Ω 2 .

Then T has a fixed point in K( Ω ¯ 2 Ω 1 ).

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

α:P[0,+)

is continuous and

α ( t x + ( 1 t ) y ) tα(x)+(1t)α(y)

for all x,yP and t[0,1].

For numbers a, b such that 0<a<b, letting α be a nonnegative continuous concave functional on P, we define the following convex sets:

P a = { x P : x < a }

and

P(α,a,b)= { x P : a α ( x ) , x b } .

Lemma 2.4 Let T: P ¯ c P ¯ c be completely continuous and α be a nonnegative continuous concave functional on P such that α(x)x for all x P ¯ c . Suppose that there exist 0<d<a<bc such that

  1. (i)

    {xP(α,a,b):α(x)>a} and α(Tx)>a for xP(α,a,b);

  2. (ii)

    Tx<d for xd;

  3. (iii)

    α(Tx)>a for xP(α,a,c) with Tx>b.

Then T has at least three fixed points x 1 , x 2 , x 3 satisfying

x 1 < d , a < α ( x 2 ) , x 3 > d and α ( x 3 ) < a .

In addition, the following compactness criterion proved by Zima in [8] is also used in our proof.

Lemma 2.5 Let ΩBC(R). Let us assume that the functions uΩ are equicontinuous in each compact interval of R and that for all uΩ, we have

| u ( x ) | ξ(x),xR,

where ξBC(R) verifies

lim | x | + ξ(x)=0.

Then Ω is relatively compact.

3 Main results

From now on, we assume that M= i j Supp( a i j )Supp( c i ) is a nonempty compact set. Let E denote the Banach space with the norm u= i = 1 n | u i | , | u i | = max x M | u i (x)| for u=( u 1 , u 2 ,, u n )E. Define a cone KE as

K= { u = ( u 1 , u 2 , , u n ) E : u i ( x ) 0  and  min x M i = 1 n u i ( x ) δ u } ,

where δ= min i = 1 , 2 , , n { m i p 0 i }(0,1), p 0 i = inf M p i (x) and the constants m i m i (M), i=1,2,,n, are defined by property (iii) of Lemma 2.1. Since M is compact, then p 0 i >0, i=1,2,,n. Moreover, from (iii) of Lemma 2.1 it follows that m i p 0 i <1 for i=1,2,,n.

Let T:KE be a map with components ( T 1 ,, T n ) defined by

T i (u)(x)= M G i (x,s) [ j i a i j u j + c i ( s ) f i ( u ( s ) ) ] ds.

A fixed point of T is a solution of (1) which belongs to .

Lemma 3.1 Assume that (H1)-(H3) hold. Then T(K)K, and T:KK is completely continuous.

Proof The continuity is trivial. Since M is compact, there exists a point x m where min x M T i (u)(x) is attained. Then, for any xR, we have

T i ( u ) ( x m ) = M G i ( x , s ) [ j i a i j u j + c i ( s ) f i ( u ( s ) ) ] d s m i M p i ( s ) G i ( s , s ) [ j i a i j u j + c i ( s ) f i ( u ( s ) ) ] d s m i p 0 i M G i ( s , s ) [ j i a i j u j + c i ( s ) f i ( u ( s ) ) ] d s m i p 0 i M G i ( x , s ) [ j i a i j u j + c i ( s ) f i ( u ( s ) ) ] d s m i p 0 i T i ( u ) ( x ) ,

namely,

min x M T i (u)(x) m i p 0 i T i (u)(x)δ T i (u)(x).

Therefore, it is clear that T(K)K.

Finally, we prove that each component of T is compact. Let ΩK be a bounded set, then there exists a constant C>0 which is uniformly bounded for its element. Since the derivative is bounded in compacts, the functions of T i (Ω) are equicontinuous on each compact interval. On the other hand, for any uΩ,

| T i ( u ) ( x ) | C R G i (x,s) ( j i a i j ( s ) ) ds+ max u C f i (u) R G i (x,s) c i (s)ds=ξ(x).

 □

Theorem 3.2 Assume that (H1)-(H3) hold. In addition, a i j (x) (ij) satisfies

M G i (s,s) ( j i a i j ) ds< 1 2 n .
  1. (a)

    If f i , 0 =0, f i 0 , = for some i 0 {1,2,,n}, then (1) has at least one positive solution.

  2. (b)

    If f i , =0, f i 0 , 0 = for some i 0 {1,2,,n}, then (1) has at least one positive solution.

Proof (a) On the one hand, since f i , 0 =0, then there exists r>0 such that

f i (u)ϵ i = 1 n u i for 0< u 1 ++ u n r,

where ϵ>0 is sufficiently small such that

ϵ M G i (s,s) c i (s)ds< 1 2 n .

Set Ω r ={uE:u<r}. Then, for any u Ω r K, we have

T i ( u ) ( x ) = M G i ( x , s ) [ j i a i j u j + c i ( s ) f i ( u ( s ) ) ] d s M G i ( s , s ) [ j i a i j ( s ) u j ( s ) + c i ( s ) ϵ i = 1 n u i ( s ) ] d s [ M G i ( s , s ) ( j i a i j ( s ) ) d s + ϵ M G i ( s , s ) c i ( s ) d s ] u 1 n u .

Furthermore, for any u Ω r K, we have

T ( u ) = i = 1 n | T i ( u ) | u.

On the other hand, since f i 0 , = for some i 0 {1,2,,n}, then there exists R ¯ such that

f i 0 (u)η i = 1 n u i for  i = 1 n u i R ¯ ,

where η>0 is sufficiently large such that

η δ 2 M G i 0 (s,s) c i 0 (s)ds>1.

Let R=max{2r, R ¯ δ } and set Ω R ={uE:u<R}. Then, for any uK Ω 2 , min x M i = 1 n u i (x)δu R ¯ , and we have

T i 0 ( u ) ( x ) = M G i 0 ( x , s ) [ j i 0 a i 0 j u j + c i 0 ( s ) f i 0 ( u ( s ) ) ] d s m i 0 p 0 i 0 M G i 0 ( s , s ) c i 0 ( s ) η i = 1 n u i ( s ) d s η δ 2 M G i 0 ( s , s ) c i 0 ( s ) d s u > u .

Furthermore, we have

T ( u ) >ufor uK Ω R .

Now by Lemma 2.3, T has a fixed point u=( u 1 , u 2 ,, u n )K( Ω ¯ R Ω r ), namely, (1) has a positive solution.

(b) On the one hand, since f i 0 , 0 = for some i 0 {1,2,,n}, there exists r>0 such that

f i 0 (u)η i = 1 n u i for 0 i = 0 n u i r,

where η>0 is sufficiently large such that

η δ 2 M G i 0 (s,s) c i 0 (s)ds>1.

Set Ω r ={uE:u<r}. Then, for any uK Ω r , we have

T i 0 ( u ) ( x ) = M G i 0 ( x , s ) [ j i 0 a i 0 j u j + c i 0 ( s ) f i 0 ( u ( s ) ) ] d s m i 0 p 0 i 0 M G i 0 ( s , s ) c i 0 ( s ) η i = 1 n u i ( s ) d s η δ 2 M G i 0 ( s , s ) c i 0 ( s ) d s u > u .

Furthermore, we have

T ( u ) >ufor uK Ω r .

On the other hand, since f i , =0, then there exists R ¯ such that

f i (u)ϵ i = 1 n u i for  u 1 ++ u n R ¯ ,

where ϵ>0 is sufficiently small such that

ϵ M G i (s,s) c i (s)ds< 1 2 n .

Let R=max{2r, R ¯ δ } and set Ω R ={uE:u<R}. Then, for any uK Ω R , min x M i = 1 n u i (x)δu R ¯ , and we have

T i ( u ) ( x ) = M G i ( x , s ) [ j i a i j u j + c i ( s ) f i ( u ( s ) ) ] d s M G i ( s , s ) [ j i a i j ( s ) u j ( s ) + c i ( s ) ϵ i = 1 n u i ( s ) ] d s [ M G i ( s , s ) ( j i a i j ( s ) ) d s + ϵ M G i ( s , s ) c i ( s ) d s ] u 1 n u .

Furthermore, for any u Ω R K, we have

T ( u ) = i = 1 n | T i ( u ) | u.

Now, by Lemma 2.3, T has a fixed point u=( u 1 , u 2 ,, u n )K( Ω ¯ R Ω r ), namely, (1) has a positive solution. □

Corollary 3.3 Assume that (H1)-(H3) hold. a i j (x) (ij) satisfies

M G i (s,s) ( j i a i j ) ds< 1 2 n .

In addition, the following conditions hold.

(H4) If there exist constants R ˆ ,ρ>0 such that for some i 0 {1,2,,n},

f i 0 (u)ρ R ˆ for δ R ˆ i = 1 n u i R ˆ ,

where ρ satisfies

ρδ M G i 0 (s,s) c i 0 (s)ds>1;

(H5) f i , 0 =0, f i , =0.

Then (1) has at least two positive solutions.

Corollary 3.4 Assume that (H1)-(H3) hold. a i j (x) (ij) satisfies

M G i (s,s) ( j i a i j ) ds< 1 2 n .

In addition, the following conditions hold.

(H6) If there exist constants R ˜ ,ϑ>0 such that

f i (u)ϑ R ˜ for δ R ˜ i = 1 n u i R ˜ ,

where ϑ satisfies

ϑ M G i (s,s) c i (s)ds< 1 2 n ;

(H7) f i 0 , 0 =, f j 0 , = for some i 0 , j 0 {1,2,,n}.

Then (1) has at least two positive solutions.

Theorem 3.5 Assume that (H1)-(H3) hold and a i j (x) (ij) satisfies

M G i (s,s) ( j i a i j ) ds< 1 2 n .

In addition, there exist numbers a, c and d with 0<d<a< c 4 such that the following conditions are satisfied:

(H8) f i (u)< 1 2 n M G i ( s , s ) c i ( s ) d s d for u i 0 and 0 i = 1 n u i <d;

(H9) there exists i 0 {1,2,,n} such that

f i 0 ( u ( x ) ) > a δ 2 Γ for xM, u i 0 and  i = 1 n u i [ a , a δ ] ,

where Γ=min{ M G i (s,s) c i (s)ds};

(H10) f i (u) 1 2 n M G i ( s , s ) c i ( s ) d s c for u i 0 and i = 1 n u i <c.

Then (1) has at least three positive solutions.

Proof For u=( u 1 , u 2 ,, u n )K, define

α(u)= min x M ( u 1 ( x ) + u 2 ( x ) + + u n ( x ) ) ,

then it is easy to know that α is a nonnegative continuous concave functional on K with α(u)u for uK.

Set b= a δ . First, we show that T: K ¯ c K ¯ c with c>b. For any u K ¯ c , we have

T i ( u ) ( x ) = R G i ( x , s ) [ j i a i j u j + c i ( s ) f i ( u ( s ) ) ] d s = M G i ( x , s ) [ j i a i j u j + c i ( s ) f i ( u ( s ) ) ] d s M G i ( s , s ) [ j i a i j u j + c i ( s ) f i ( u ( s ) ) ] d s M G i ( s , s ) [ j i a i j u j + c i ( s ) c 2 n M G i ( s , s ) c i ( s ) d s ] d s c 2 n + c 2 n = c n .

So T(u)= i = 1 n | T i ( u ) | c.

In a similar way, we also can prove that T: K ¯ d K d . Then (ii) of Lemma 2.4 holds.

Next, we shall show that (i) of Lemma 2.4 is satisfied. It is clearly seen that u=( a + b 2 n ,, a + b 2 n ){u=( u 1 , u 2 ,, u n )K(α,a,b):α(u)>a}. Then, for any uK(α,a,b) and xM, it is easy to obtain that

b i = 1 n | u i | i = 1 n u i (x) min x M ( i = 1 n u i ( x ) ) =α(u)>a.

Then, by (H9), we can have

α ( T ( u ) ( x ) ) = min x M ( i = 1 n T i ( u ) ( x ) ) min x M T i 0 ( u ) ( x ) δ max x M T i 0 ( u ) ( x ) δ M G i 0 ( x , s ) [ j i 0 a i 0 j u j + c i 0 ( s ) f i 0 ( u ( s ) ) ] d s δ M m i 0 p i 0 ( s ) G i 0 ( s , s ) c i 0 ( s ) f i 0 ( u ( s ) ) d s δ 2 M G i 0 ( s , s ) c i 0 ( s ) a δ 2 Γ d s = a .

Finally, we verify that (iii) of Lemma 2.4 is satisfied. Suppose that uK(α,a,c) with T(u)>b, then we can have

α ( T ( u ) ) = min M ( i = 1 n T i ( u ) ( x ) ) δ T ( u ) > b δ = a .

From the above, the hypotheses of Leggett-Williams theorem are satisfied. Hence (1) has at least three positive solutions u 1 , u 2 and u 3 such that u 1 <d, a< min x M ( i = 1 n u i 2 (x)), and u 3 >d with

min x M ( i = 1 n u i 3 ( x ) ) <a.

 □

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Correspondence to Fanglei Wang.

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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.

An erratum to this article is available at http://dx.doi.org/10.1186/s13661-016-0714-4.

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Keywords

  • nonlinear Schrödinger systems
  • positive solutions
  • fixed point theorems