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A Liouville-type theorem for the coupled Schrödinger systems and the uniqueness of the sign-changing radial solutions

Haoyu Li, Olímpio Hiroshi Miyagaki

Abstract

In this paper, we study the sign-changing radial solutions of the following coupled Schrödinger system \begin{equation} \left\{ \begin{array}{lr} -Δu_j+λ_j u_j=μ_j u_j^3+\sum_{i\neq j}β_{ij} u_i^2 u_j \,\,\,\,\,\,\,\, \mbox{in }B_1 ,\nonumber u_j\in H_{0,r}^1(B_1)\mbox{ for }j=1,\cdots,N.\nonumber \end{array} \right. \end{equation} Here, $λ_j,\,μ_j>0$ and $β_{ij}=β_{ji}$ are constants for $i,j=1,\cdots,N$ and $i\neq j$. $B_1$ denotes the unit ball in the Euclidean space $\mathbb{R}^3$ centred at the origin. For any $P_1,\cdots,P_N\in\mathbb{N}$, we prove the uniqueness of the radial solution $(u_1,\cdots,u_j)$ with $u_j$ changes its sign exactly $P_j$ times for any $j=1,\cdots,N$ in the following case: $λ_j\geq1$ and $|β_{ij}|$ are small for $i,j=1,\cdots,N$ and $i\neq j$. New Liouville-type theorems and boundedness results are established for this purpose.

A Liouville-type theorem for the coupled Schrödinger systems and the uniqueness of the sign-changing radial solutions

Abstract

In this paper, we study the sign-changing radial solutions of the following coupled Schrödinger system \begin{equation} \left\{ \begin{array}{lr} -Δu_j+λ_j u_j=μ_j u_j^3+\sum_{i\neq j}β_{ij} u_i^2 u_j \,\,\,\,\,\,\,\, \mbox{in }B_1 ,\nonumber u_j\in H_{0,r}^1(B_1)\mbox{ for }j=1,\cdots,N.\nonumber \end{array} \right. \end{equation} Here, and are constants for and . denotes the unit ball in the Euclidean space centred at the origin. For any , we prove the uniqueness of the radial solution with changes its sign exactly times for any in the following case: and are small for and . New Liouville-type theorems and boundedness results are established for this purpose.
Paper Structure (11 sections, 13 theorems, 36 equations)

This paper contains 11 sections, 13 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Main theorems
  3. Some historical remarks
  4. The idea of the prove and the organization of this note
  5. Preliminaries
  6. Notations
  7. Liouville-type theorems and a priori estimates
  8. Tanaka2016 and the corresponding non-degeneracy result
  9. Proof of Theorem \ref{['t:main']} and Theorem \ref{['t:boundedness']}
  10. Proof of Theorem \ref{['t:boundedness']} and its corollaries
  11. Proof of Theorem \ref{['t:main']}

Key Result

Theorem 1.1

If $\lambda_1,\cdots,\lambda_N\geq1$, for any $P_1,\cdots,P_N\in\mathbb{N}$, there exists a positive number $b=b(\lambda_1,\cdots,\lambda_N;\mu_1,\cdots,\mu_N;P_1,\cdots,P_N)>0$ such that if $|\beta_{ij}|<b$, Problem (e:AAAA) admits an unique solution $(u_1,\cdots,u_N)$ with $u_j(0)>0$ and $u_j$ cha

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Corollary 2.5
  • Corollary 2.6
  • ...and 11 more