Spike vector solutions for some coupled nonlinear Schrödinger equations

Shuangjie Peng; Huirong  Pi

doi:10.3934/dcds.2016.36.2205

Article Contents

2016, Volume 36, Issue 4: 2205-2227. Doi: 10.3934/dcds.2016.36.2205

This issue Previous Article Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge Next Article On the Cauchy problem for the Boltzmann equation in Chemin-Lerner type spaces

Spike vector solutions for some coupled nonlinear Schrödinger equations

Shuangjie Peng^1, and
Huirong Pi^2,

1.
Department of Mathematics, Central China Normal University, Wuhan 430079
2.
Center for Partial Differential Equations, East China Normal University, Shanghai, 200241

Received: January 31, 2014

Revised: June 30, 2015

Published: May 2017

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Abstract

We consider spike vector solutions for the nonlinear Schrödinger system \begin{equation*} \left\{ \begin{array}{ll} -\varepsilon^{2}\Delta u+P(x)u=\mu u^{3}+\beta v^2u \ \hbox{in}\ \mathbb{R}^3,\\ -\varepsilon^{2}\Delta v+Q(x)v=\nu v^{3} +\beta u^2v \ \ \hbox{in}\ \mathbb{R}^3,\\ u, v >0 \,\ \hbox{in}\ \mathbb{R}^3, \end{array} \right. \end{equation*} where $\varepsilon > 0$ is a small parameter, $P(x)$ and $Q(x)$ are positive potentials, $\mu>0, \nu>0$ are positive constants and $\beta\neq 0$ is a coupling constant. We investigate the effect of potentials and the nonlinear coupling on the solution structure. For any positive integer $k\ge 2$, we construct $k$ interacting spikes concentrating near the local maximum point $x_{0}$ of $P(x)$ and $Q(x)$ when $P(x_{0})=Q(x_{0})$ in the attractive case. In contrast, for any two positive integers $k\ge 2$ and $m\ge 2$, we construct $k$ 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 $\bar{x}_{0}$ of $Q(x)$ respectively when $x_{0}\neq \bar{x}_{0}$, moreover, spikes of $u$ and $v$ repel each other. Meanwhile, we prove the attractive phenomenon for $\beta < 0$ and the repulsive phenomenon for $\beta > 0$.

Keywords:

Coupled nonlinear Schrödinger equations,
asymptotic behavior,
spike vector solutions,
Lyapunov-Schmidt reduction,
critical point.

Mathematics Subject Classification: Primary: 35B40; Secondary: 35B20, 35J47, 35Q40.

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