Multi-Peak Solutions for Coupled Nonlinear Schrödinger Systems in Low Dimensions

In this paper, we construct the solutions to the following nonlinear Schrödinger system -ϵ2Δu+P(x)u=μ1up+βup-12vp+12inRN,-ϵ2Δv+Q(x)v=μ2vp+βup+12vp-12inRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\epsilon ^{2}\Delta u+P(x)u= \mu _{1} u^{p}+\beta u^{\frac{p-1}{2}}v^{\frac{p+1}{2}} \ \ \ \text {in} \ \ \mathbb {R}^{N},\\ -\epsilon ^{2}\Delta v+Q(x)v= \mu _{2} v^{p}+\beta u^{\frac{p+1}{2}}v^{\frac{p-1}{2}} \ \ \ \text {in} \ \ \mathbb {R}^{N}, \end{array}\right. } \end{aligned}$$\end{document}where 3<p<+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3< p<+\infty $$\end{document}, N∈{1,2}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in \{1,2\}$$\end{document}, ϵ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon >0$$\end{document} is a small parameter, the potentials P, Q satisfy 0<P0≤P(x)≤P1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<P_{0} \le P(x)\le P_{1}$$\end{document} and Q(x) satisfies 0<Q0≤Q(x)≤Q1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<Q_{0} \le Q(x)\le Q_{1}$$\end{document}. We construct the solution for attractive and repulsive cases. When x0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{0}$$\end{document} is a local maximum point of the potentials P and Q and P(x0)=Q(x0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(x_{0})=Q(x_{0})$$\end{document}, we construct k spikes concentrating near the local maximum point x0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{0}$$\end{document}. When x0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{0}$$\end{document} is a local maximum point of P and x¯0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{x}_{0}$$\end{document} is a local maximum point of Q, we construct k spikes of u concentrating at the local maximum point x0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{0}$$\end{document} and m spikes of v concentrating at the local maximum point x¯0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{x}_{0}$$\end{document} when x0≠x¯0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{0}\ne \overline{x}_{0}.$$\end{document} This paper extends the main results established by Peng and Wang (Arch Ration Mech Anal 208:305–339, 2013) and Peng and Pi (Discrete Contin Dyn Syst 36:2205–2227, 2016), where the authors considered the case N=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=3$$\end{document}, p=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=3$$\end{document}.

where 3 < p < +∞, N ∈ {1, 2}, > 0 is a small parameter, the potentials P, Q satisfy 0 < P 0 ≤ P(x) ≤ P 1 and Q(x) satisfies 0 < Q 0 ≤ Q(x) ≤ Q 1 . We construct the solution for attractive and repulsive cases. When x 0 is a local maximum point of the potentials P and Q and P(x 0 ) = Q(x 0 ), we construct k spikes concentrating near the local maximum point x 0 . When x 0 is a local maximum point of P and x 0 is a local maximum point of Q, we construct k spikes of u concentrating at the local maximum point x 0 and m spikes of v concentrating at the local maximum point x 0 when x 0 = x 0 . This paper extends the main results established by Peng

Introduction and Main Results
In this paper, we construct the solutions for the following nonlinear Schrödinger system 1 2 in R N , where > 0 is a small parameter, 5 ≤ p < +∞, N ∈ {1, 2}, and the potentials P, Q satisfy 0 < P 0 ≤ P(x) ≤ P 1 , Q(x), respectively 0 < Q 0 ≤ Q(x) ≤ Q 1 . The use of the Lyapunov-Schmidt reduction method to construct solutions for the nonlinear Schrödinger equation attracted much attention in the last decade, starting from the pioneering contribution by Floer and Weinstein [11]. Noussair and Yan [21] considered multi-peak solutions for the following problem when x 0 is a local maximum point of Q(x) and is sufficiently small, they proved that for each positive integer k, problem (1.2) has a positive solution with k-peaks concentrating near x 0 . Wei and Yan [32] studied the following nonlinear Schroödinger equation They proved that problem (1.3) has infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. For more results about the nonlinear Schrödinger equation, we refer the reader to [1,6,[8][9][10]12] and the references therein.
(H1) (Q) There are constants b ∈ R, n > 1, and > 0 such that as r → +∞ By using the number of the bumps of the solutions as a parameter, for the repulsive case, they constructed non-radial positive vector solutions of segregated type, for the attractive case, they constructed non-radial positive vector solutions of synchronized type. When N = 3, p = 3, Peng and Pi [22] constructed k interacting spikes for u near the local maximum point x 0 of P(x) and m interacting spikes for v near the local maximum point x 0 of Q(x), respectively, when x 0 = x 0 . Tang and Xie in [27] constructed synchronized positive vector solutions for small. Since it seems to be difficult to provide a complete list of references, we just refer the readers to [2-4, 7, 13, 14, 17-20, 22-26, 28-30, 33, 34] and the references therein.
In this paper, we have been inspired by the analysis developed in [21-23, 31, 32] for scalar nonlinear elliptic equations (systems), in particular, by the ideas introduced by Noussair and Yan [21] to deal with nonlinear elliptic equations. Compared with the single scalar equation, we encounter some new difficulties in estimates due to the nonlinear coupling. Firstly, we need to establish non-degenerate results for the solutions of the coupled system, which will be used to prove the invertibility of the operator for the repulsive case. The difficulty is that we need to give an exact integral estimate for the coupled term and the system is more complicated than single equations. We point out that the sign of β has great influence on the structure of the solutions. Roughly speaking, for the repulsive case, the solutions are small perturbations of (U , V ), where (U , V ) are scaling and translation of the solution of − u + λu = u p . For the attractive case, the solutions are small perturbations of (U μ , U ν ), where U μ are scaling and translation of the solution of − u + λu = μu p and U ν are scaling and translation of the solution of − u + λu = νu p .
In this paper, we examine how potentials and the interspecies scattering length β influence the structure of solutions to problem (1.1), which improves the results of [19,22] for least energy solutions. We study the existence of high energy solutions to problem (1.1) and provide not only the locations of spikes, but also much finer information on the interaction of spikes. Furthermore, we prove the attractive phenomenon for β < 0 and the repulsive phenomenon for β > 0.

Consider the following system
− u + λu = μ 1 u p + βu 1 2 in R N . (1.4) Let W be the unique solution of is a solution of (1.4), where λ := P(x 0 ) = Q(x 0 ) and τ 0 satisfies In the sequel, we will use U ,x j, (x), V ,x j, (x) to build up the solutions of (1.1).
To show our main results, we first recall some known results from [15]. In the following, let

Remark 1.1
Although the authors in [15] dealt with the existence and the nondegeneracy of positive solutions for the fractional Schrödinger system, the main results are still true for the classical Schrödinger system with s = 1.
The main results in this paper are stated in what follows.
We will use U ,x j ,μ (x), U ,z j ,ν (x) to build up the solutions of problem (1.1).

Theorem 1.2
Suppose that x 0 is a local maximum point of P(x) and x 0 is a local maximum point of Q(x), with x 0 = x 0 . Then there exists β * > 0 depending on x 0 and x 0 such that for all β < β * , there exists 0 > 0 such that for any ∈ (0, 0 ], problem (1.1) has a solution of the form

Remark 1.2
The conditions in Theorem 1.1 ensure the existence and the nondegeneracy of positive solutions (U , V ) to the related limit problem (1.4), hence the reduction procedure can be carried out successfully. The condition β < β * in Theorem 1.2 is necessary to prove that Q B (see (2.5)) is invertible and the inverse operator is bounded.

Remark 1.3
Compared with the well-studied case N = 3, p = 3, in order to get an accurate error estimate and use the Contraction Mapping Theorem to prove that problem (2.54) has a unique solution, our situation is much more complicated. In this sense, we complement the main results established by Peng and Wang (Arch Ration Mech Anal 2013) and Peng and Pi (Discrete Contin Dyn Syst 2016), where the authors considered the case N = 3, p = 3.
The paper is organized as follows. In Sect. 2, we introduce some preliminaries that will be used to prove Theorems 1.1-1.2. In Sect. 3, we prove Theorem 1.1. In Sect. 4, we prove Theorem 1.2. Finally, we give some elementary computations in Appendix A.

Preliminary Results
We first give the definition of multi-peak solutions of system (1.1).
(ii) For any given τ > 0, there exists R 1, such that (iii) There exists C > 0 such that Let x 0 be the local maximum points of P(x), Q(x) and P(x 0 ) = Q(x 0 ). We want to construct a solution (u , v ) of the following form Then, (ϕ , ψ ) satisfies the following equation From [8,9], we have following estimates.
to E as follows: Therefore b ,i, j is determined by the following equations: We now prove that problem (2.6) is solvable. (2.8) By Lemma 2.1, (2.8) and the symmetry of U ,x j, , V ,x j, , we have where δ h,i = 0 if h = i and δ i,i = 1, c j > 0 is a constant. Hence (2.6) is solvable and we have the following estimate In order to prove the invertibility of the operator Q B , we use the following non-degenerate results for system (1.4), which can be found in [15].
in the sense that the kernel is given by span In order to carry out the reduction arguments, we give following key lemma. (2.14) By Hölder's inequality and Young's inequality, we have On the other hand, we can take a large R > 0 such that If we can prove that we get a contradiction. For this purpose, we will discuss the local behaviors near each points x n ,m . We define Therefore, To prove (2.20), we only need to show that (ϕ, ψ) = (0, 0). Remark that relation By (2.27) and (ϕ n , ψ n ) ∈ E n , we can estimate γ n,i, j as following On the other hand, from (2.8) and (ϕ n , ψ n ) ∈ E n , we have Combining (2.28), (2.29) with above A 1 to A 6 , we have (2.38) From proposition 2.1, the solution of (U λ , V λ ) gives On the other hand, from (ϕ n , ψ n ) ∈ E n and (2.29), we have So, we have prove there exist 0 , θ 0 > 0, ρ > 0, independent of x j , j = 1, 2, · · · k such that for any ∈ (0, 0 ] and Thus the proof is complete. Next, we give the error estimate for l and R (ϕ , ψ ) .

Lemma 2.3
There is a constant C > 0 independent of , such that Firstly, by Hölder's inequality, we have (2.47) So, by (2.42) and (2.47), we obtain This completes the proof.

Lemma 2.4
There is a constant C > 0 independent of , such that where For any ξ, we let ξ(y) = ξ( y), then When p ≥ 3, by (2.49) and the Hölder inequality, we have Similarly, we have (2.51) Thus We expand B 3 as following Since V ,x j, = τ 0 U ,x j, , we have h) .
By the similar argument as C 3 , we have We also have Combining (2.50)-(2.53), the estimates for C 1 -C 6 and (2.48), we obtain This finishes the proof. Now, we consider the following projection problem by using the contraction mapping theorem, we give the following lemma.

By Lemmas 2.2 and 2.3, we have
(2.55) Next, we will use the contraction mapping theorem in a ball whose radius is slightly bigger than C l . So we take where τ > 0 is a fixed small constant.
Step 1 B is a map from S to S. In fact, from Lemmas 2.2, 2.3 and 2.4, we have Thus, B is a map from S to S.

Moreover, from Lemma 2.3 and Lemma 2.4, we have
As desired.
Next, we solve equation (2.1). Since From Lemma 2.5, we know the following equation has a unique solution (ϕ , ψ ). So Firstly, it is easy to see that the right hand of (2.65) belongs to E , if the left hand of of (2.65) belongs to E , then the right hand of (2.65) must be zero. Let

Proof of Theorem 1.1
In this section, we give the proof of Theorem 1.1.
Proof In order to solve (2.65), we define a function as following Then, from Lemmas 2.3, 2.4 and Proposition 5.1, there exists a small constant σ > 0 such that We use the ideas introduced in [22] to consider the following maximizing problem We prove that if max x∈D K (x) is achieved by some x in D, then x is an interior point of D. Taking Therefore, x is an interior point of D, which implies that is a critical point of K (x). So, (1.1) has a solution of the form ).

Proof of Theorem 1.2
This section is devoted to the proof of Theorem 1.2.
Let λ j = P(x j ), λ j = Q(z j ) and x 0 is a local maximum point of P(x) and x 0 is a local maximum point of Q(x), we want to construct a solution (u , v ) of the following form Then, ( ϕ , ψ ) satisfies the following equation where B is a bounded linear operator in Lemma 4.1 There exist β * > 0 and 0 , θ 0 > 0, ρ > 0, independent of x j , j = 1, 2, . . . k and z j , j = 1, 2, . . . m, such that for any ∈ (0, 0 ], x j ∈ B θ 0 (x 0 ) and Proof The proof is similar as the proof of Lemma 2.2. To get a contradiction, we take the projection to We need to prove ϕ = 0 and ψ = 0. When we prove ϕ = 0, we only need to set h = 0, ϕ n = 0 after (2.23) in Lemma 2.2. So we will get ϕ satisfies − ϕ + λϕ − pμU p−1 r = 0. By the non-degenerate of the solution of − u + λu = μu p , we Then, we can get c i = 0, thus ϕ = 0. To show ψ = 0, we need to set g = 0, ψ n = 0 after (2.23) in Lemma 2.2. So we will get that ψ satisfies − ψ + λψ − pνU p−1 r = 0. By the non-degenerate of the solution of ∂ y i . Then, we can get d i = 0, thus ψ = 0. The rest is similar to the proof of (3.8) in Lemma 3.2 in [23], we can get a contradiction when β < β * .
To carry out the contraction mapping theorem, we give the following error estimate.

Lemma 4.2 There is a constant C
Proof The proof is similar to that of Lemma 2.3. First, by the Hölder inequality, we have h) .
This proof is thus complete.

Lemma 4.3
There is a constant C > 0 independent of and a small constant σ > 0 such that Proof The proof is similar to that of Lemma 2.4, we only need to small changes, so, we omit the details here.
Based on the similar arguments as in Lemma 2.5, we have the following lemma.

Lemma 4.4
There exists 0 > 0, such that for any ∈ (0, 0 ], x j ∈ B θ (x 0 ), z j ∈ B θ (x 0 ), then (2.54) has a unique (ϕ , ψ ) ∈ E and Proof of Theorem 1.2 In order to solve (2.65), we define a function as following Consider the following maximizing problem where Thus, we can get Data Availability This paper has no associated data and material.

Conflict of interest
The authors have no relevant financial or non-financial interests to disclose.
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By the definition of U ,x j, and V ,x j, , we get From (2.42), we know , by the similar argument as (2.43)-(2.45) we have there exists a σ > 0 such that  Thus, ).

Proposition 5.2
There exist positive constants C 1 , C 2 , C 3 and C 4 , such that Proof By direct computations, we have Thus, by the above computations, we obtain the desired conclusions.