Multiple solutions for a nonlinear Schrödinger systems

Fengshuang Gao; Yuxia Guo

doi:10.3934/cpaa.2020055

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2020, Volume 19, Issue 2: 1181-1204. Doi: 10.3934/cpaa.2020055

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Multiple solutions for a nonlinear Schrödinger systems

Fengshuang Gao and
Yuxia Guo^,

Department of Mathematics, Tsinghua University, Beijing, 100084, China

^* Corresponding author
^* Corresponding author

Received: April 30, 2019

Revised: April 30, 2019

Published: October 2019

This work is supported by NSFC (11571040, 11771235)

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Abstract

This paper is concerned with the following nonlinear Schrödinger systems:

$ \left\{\begin{aligned}-\Delta u+a(x)u& = |u|^{p-2}u+\beta |u|^{\frac{p}{2}-2}u|v|^{\frac{p}{2}} \ \ \hbox{in } \mathbb{R}^{N}\\-\Delta v+a(x)v& = |v|^{p-2}v+\beta |v|^{\frac{p}{2}-2}v|u|^{\frac{p}{2}}\ \ \ \hbox{in } \mathbb{R}^{N}\\(u,v)&\in (H^1(\mathbb{R}^N))^2,\end{aligned}\right. \ \ \ \ \ \ \ {(P)} $

where $ N\geq3 $ and $ 2<p<\frac{2N}{N-2} = 2^{\ast} $, $ \beta\in \mathbb{R} $ is a coupling constant. $ a(x) $ is a $ \mathcal{C}^1 $ potential function. In the repulsive case, i.e. $ \beta<0 $, under some suitable decay assumptions but without any symmetric assumptions on the potential $ a(x) $, we prove the existence of infinitely many solutions for the problem $ (P). $

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Mathematics Subject Classification: Primary: 35B45, 35J25.

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