Multi-spikes solutions for a system of coupled elliptic equations with quadratic nonlinearity

Zhongwei Tang; Huafei Xie

doi:10.3934/cpaa.2020017

Article Contents

2020, Volume 19, Issue 1: 311-328. Doi: 10.3934/cpaa.2020017

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Multi-spikes solutions for a system of coupled elliptic equations with quadratic nonlinearity

Zhongwei Tang and
Huafei Xie^,

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

^* Corresponding author
^* Corresponding author

Received: November 30, 2018

Revised: February 28, 2019

Published: July 2019

The first author is supported by NSFC (11571040, 11671331)

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Abstract

This paper is devoted to study the following systems of coupled elliptic equations with quadratic nonlinearity

$ \begin{equation*} \begin{cases} -\varepsilon^{2}\Delta v+P(x)v = \mu vw, &x\in{\mathbb{R}}^{N},\\ -\varepsilon^{2}\Delta w+Q(x)w = \frac{\mu}{2} v^{2}+\gamma w^{2}, &x\in{\mathbb{R}}^{N}, \end{cases} \end{equation*} $

which arises from second- harmonic generation in quadratic optical media. We assume that the potential functions $ P(x) $ and $ Q(x) $ are positive functions and have a strict local maxima at $ x_{0} $. Applying the finite dimensional reduction method, for any integer $ 1\leq k\leq N+1 $, we prove the existence of positive solutions which have $ k $ local maximum points that concentrate at $ x_{0} $ simultaneously when $ \varepsilon $ is small.

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Mathematics Subject Classification: Primary: 35J60; Secondary: 35J20.

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