Multi-Peak Positive Solutions for Nonlinear Fractional Schr\"{o}dinger Systems in ℝ^N

1 Introduction

In this paper, we are concerned with the following nonlinear fractional Schrödinger system:

where ϵ>0 is a small parameter, μ1,μ2>0 and β>0 is a coupling constant, P⁢(x) and Q⁢(x) are potential functions, N>2⁢s, 0<s<1, 2<p<2*⁢(s)2, 2*⁢(s)=2⁢NN-2⁢s.

Problem (1.1) arises from looking for standing waves (ψ(t,x),φ(t,x))=(exp(iEt)u(x), exp(iEt)v(x)) for the following nonlinear Schrödinger system:

{ i ⁢ ∂ ⁡ ψ ∂ ⁡ t = ϵ 2 ⁢ s ⁢ ( - Δ ) s ⁢ ψ + ( P ⁢ ( x ) - E ) ⁢ ψ - μ 1 ⁢ | ψ | 2 ⁢ p - 2 ⁢ ψ - β ⁢ | φ | p ⁢ ψ p - 2 ⁢ ψ , x ∈ ℝ N , t > 0 , i ⁢ ∂ ⁡ φ ∂ ⁡ t = ϵ 2 ⁢ s ⁢ ( - Δ ) s ⁢ φ + ( Q ⁢ ( x ) - E ) ⁢ φ - μ 2 ⁢ | φ | 2 ⁢ p - 2 ⁢ φ - β ⁢ | ψ | p ⁢ φ p - 2 ⁢ φ , x ∈ ℝ N , t > 0 ,

where i is the imaginary unit, the constants μ1 and μ2 represent the intraspecies scattering lengths and β is the interspecies scattering length. The sign of the interspecies scattering length determines whether the interaction of states is repulsive or attractive.

It is known, but not completely trivial, that (-Δ)s reduces to the standard Laplacian -Δ as s→1. When s=1, problem (1.1) arises in various applications, such as nonlinear optics, plasma physics and condensed matter physics. The related problems have attracted considerable attention in recent years, and there are several results in the literature on the existence of such solutions. No matter the interspecies scattering length β is positive or negative, Lin and Wei [18] obtained least energy solutions for the two coupled nonlinear Schrödinger systems with the trap potentials by using Nehari’s manifold and derived their asymptotic behaviors by some techniques of singular perturbation problem. Chen, Lin and Wei [10] proved the existence of the positive solutions with any prescribed spikes by the reduction methods. Alves [1] dealt with the existence and the concentration of positive solutions by the mountain pass theorem. Wan and Ávila [27] utilized the category theory studying the relation between the number of positive standing waves solutions for the special system (1.1) with P⁢(x)=Q⁢(x) and β=0 and the topology of the set of minimum points of potentials. Pomponio [24] also proved the existence of concentrating solutions for a general system with repulsive interaction of states and that how the location of the concentration points depends strictly on the potentials. Peng and Wang [23] applied the finite reduction method to construct an unbounded sequence of non-radial positive vector solutions of segregated type in repulsive case and synchronized type in attractive case for (1.1) with p=2, N=3 for fixed ϵ. For more related results, one can refer to [6, 29, 28, 2, 3, 4, 5, 16, 20, 19] and the references therein.

When 0<s<1, the single equation

has been extensively investigated based on various assumptions on the potential V. Chen and Zheng [9] showed that if N=1,2,3, ϵ is sufficiently small, max⁡{12,N4}<s<1 and V satisfies some smoothness and boundedness assumptions, then equation (1.2) has a nontrivial solution uϵ concentrated to some single point as ϵ→0. Dávila, del Pino and Wei [11] generalized various existence results known for (1.2) with s=1 to the case of fractional Laplacian. Long, Peng and Yang [21] showed that (1.2) has infinitely many non-radial positive solutions and sign-changing solutions, whose energy can be made arbitrarily large. For more related results, one can refer to [7, 13, 25, 26] and the references therein.

In this paper, inspired by [21, 22], we intend to prove that the nonlinear fractional Schrödinger system (1.1) has multi-peak solutions. In [22], a positive solution with m-peaks concentrating near x0 was found for (1.2) with s=1. Compared with the operator -Δ, which is local, the operator (-Δ)s with 0<s<1 on ℝN is nonlocal. Unlike the local case s=1, the leading order of the associated reduced functional in a variational reduction procedure is of polynomial instead of exponential order, due to the nonlocal effect. So we need to establish some new necessary estimates for the Lyapunov–Schmidt reduction.

As far as we know, there are no results on the existence of multi-peak solutions to the nonlinear fractional Schrödinger system. In this paper, we will present some results which contribute to this respect.

We consider the problem

where P⁢(x) and Q⁢(x) are bounded positive continuous functions on ℝN satisfying the following assumptions:

1. P has a strict local maximum at some point x0∈ℝN, that is, for some δ>0, one has P⁢(x)<P⁢(x0) for all 0<|x-x0|<δ.

2. Q has a strict local maximum at some point x0∈ℝN, that is, for some δ>0, one has Q⁢(x)<Q⁢(x0) for all 0<|x-x0|<δ.

3. infx∈ℝN⁡P⁢(x)>0, infx∈ℝN⁡Q⁢(x)>0, and for all y,z∈ℝN, there exist positive constants C and θ such that |P⁢(y)-P⁢(z)|≤C⁢|y-z|θ and |Q⁢(y)-Q⁢(z)|≤C⁢|y-z|θ.

Making the change of variable x↦ϵ⁢x, we can rewrite (1.3) as the following equation:

The norm of Hs⁢(ℝN) is defined by

∥ u ∥ s = 〈 u , u 〉 , u ∈ H s ⁢ ( ℝ N ) ,

where

For any function K⁢(x)>0, we use ∥⋅∥s,ϵ,K to denote the norm in Hϵ,Ks⁢(ℝN), that is to say

∥ u ∥ s , ϵ , K = ( ∫ ℝ N ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y + K ⁢ ( ϵ ⁢ x ) ⁢ u 2 ⁢ d ⁢ x ) 1 2   for all ⁢ u ∈ H ϵ , K s ⁢ ( ℝ N ) .

Define Hϵs to be the product space Hϵ,Ps⁢(ℝN)×Hϵ,Qs⁢(ℝN) with the norm

∥ ( u , v ) ∥ s , ϵ 2 = ∥ u ∥ s , ϵ , P 2 + ∥ v ∥ s , ϵ , Q 2 .

Denote the solution of the following problem by w:

It is well known that the unique solution w of (1.5) satisfies w⁢(x)=w⁢(|x|) and w′<0 (see [14, 15]).

Note that

w λ j , μ j = ( λ j μ j ) 1 2 ⁢ ( p - 1 ) ⁢ w ⁢ ( λ j 1 2 ⁢ s ⁢ x )

solves

{ ( - Δ ) s ⁢ u + λ j ⁢ u = μ j ⁢ u 2 ⁢ p - 1 , u > 0 in ℝ N , u ⁢ ( 0 ) = max x ∈ ℝ N ⁡ u ⁢ ( x ) ,

where j=1,2 and λ1=P⁢(x0), λ2=Q⁢(x0). Denote

Let Br={x∈ℝN:|x|<r}, and B¯r be its closure. For δ>0, and any positive integers l and m, we define

Denote

E ϵ , l , m = { ( φ , ψ ) ∈ H ϵ s : 〈 φ , ∂ ⁡ u ϵ , μ 1 , y j ∂ ⁡ y i j 〉 s , ϵ , P = 〈 ψ , ∂ ⁡ v ϵ , μ 2 , z k ∂ ⁡ z i k 〉 s , ϵ , Q = 0 , j = 1 , … , l , k = 1 , … , m } ,

where i=1,2,…,N. Set

M ϵ , l , m = { ( 𝐲 , 𝐳 , φ , ψ ) : ( 𝐲 , 𝐳 ) ∈ D ϵ , δ l , m , ( φ , ψ ) ∈ E ϵ , l , m } .

The main result of this paper is as follows.

Our paper is organized as follows. In Section 2, we will give some preliminaries. In Section 3, we do the finite reduction. In Section 4, we prove our main result. Some technical estimates are left in Appendix A.

2 Some Preliminaries

In this section, we recall some properties of the fractional order Sobolev spaces and give some results which are crucial in our proof of the main theorem.

Let 0<s<1. Various definitions of the fractional Laplacian (-Δ)s⁢f⁢(x) of a function f defined in ℝN are available, depending on its regularity and growth properties.

The fractional Laplacian (-Δ)s⁢f of a function f∈Hs⁢(ℝN) is defined in terms of its Fourier transform by

( - Δ ) s ^ ⁢ f ⁢ ( ξ ) = | ξ | 2 ⁢ s ⁢ f ^ ⁢ ( ξ ) ,

where ^ is the Fourier transform. When f has some sufficiently regularity, the fractional Laplacian of a function f:ℝN→ℝ is expressed by the formula

( - Δ ) s ⁢ f ⁢ ( x ) = C N , s ⁢ P . V . ∫ ℝ N f ⁢ ( x ) - f ⁢ ( y ) | x - y | N + 2 ⁢ s ⁢ 𝑑 y = C N , s ⁢ lim ϵ → 0 ⁡ ∫ ℝ N ∖ B ϵ ⁢ ( x ) f ⁢ ( x ) - f ⁢ ( y ) | x - y | N + 2 ⁢ s ⁢ 𝑑 y ,

where CN,s=π-(2⁢s+N/2)⁢Γ⁢(N/2+s)Γ⁢(-s). This integral makes sense directly when s<12 and f∈C0,α⁢(ℝN) with α>2⁢s, or if f∈C1,α⁢(ℝN) with 1+2⁢α>2⁢s. It is well known that (-Δ)s on ℝN with 0<s<1 is a nonlocal operator.

When s∈(0,1), the space Hs⁢(ℝN)=Ws,2⁢(ℝN) is defined by

and the norm is

∥ u ∥ s = ( ∫ ℝ N ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y + ∫ ℝ N | u | 2 ⁢ 𝑑 x ) 1 2 .

Here the term

[ u ] s := ( ∫ ℝ N ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | 2 | x - y | N + 2 ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y ) 1 2

is the so-called Gagliardo (semi-)norm of u. The following identity yields the relation between the fractional operator (-Δ)s and the fractional Sobolev space Hs⁢(ℝN):

[ u ] s = C ⁢ ( ∫ ℝ N | ξ | 2 ⁢ s ⁢ | u ^ ⁢ ( ξ ) | 2 ⁢ 𝑑 ξ ) 1 2 = C ⁢ ∥ ( - Δ ) s 2 ⁢ u ∥ L 2 ⁢ ( ℝ N )

for a suitable positive constant C depending only on s and N.

Now, we recall some known results for the limit equation (1.5). If s=1, the uniqueness and non-degeneracy of the ground state U for (1.5) are due to [17]. In the celebrated paper [14], Frank and Lenzmann proved the uniqueness of ground state solution U⁢(x)=U⁢(|x|)≥0 for N=1,0<s<1,1<p<2*⁢(s)-1. Recently, Frank, Lenzmann and Silvestre [15] obtained the non-degeneracy of ground state solutions for (1.5) in arbitrary dimension N≥1 and any admissible exponent 1<p<2*⁢(s)-1.

For convenience, we summarize the properties of the ground state U of (1.5) which can be found in [14, 15].

By [15, Lemma C.2], ∂xj⁡U has the following decay estimate for j=1,…,N:

| ∂ x j ⁡ U | ≤ C 1 + | x | N + 2 ⁢ s .

From Lemma 2.2, we see the ground bound state solution as 1|x|N+2⁢s when |x|→∞. Fortunately, this polynomial decay is enough for us in the estimates of our proof.

3 Variational Reduction

In this section, we will do the finite reduction.

Define

It is easy to check that

is a bounded bi-linear functional in Eϵ,l,m. Hence, by the Lax–Milgram theorem there is a bounded linear operator ℒ from Eϵ,l,m to Eϵ,l,m such that

The following result implies that ℒ is invertible in Eϵ,l,m.