Infinitely many segregated solutions for coupled nonlinear Schrödinger systems

Lushun Wang; Minbo Yang; Yu Zheng

doi:10.3934/dcds.2019265

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2019, Volume 39, Issue 10: 6069-6102. Doi: 10.3934/dcds.2019265

This issue Previous Article Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain Next Article Standing waves for Schrödinger-Poisson system with general nonlinearity

Infinitely many segregated solutions for coupled nonlinear Schrödinger systems

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

^* Corresponding author: Lushun Wang
^* Corresponding author: Lushun Wang

Received: January 31, 2019

Revised: March 31, 2019

Published: July 2019

The second author is partially supported by NSFC (11571317, 11671364) and ZJNSF (LD19A010001)

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Abstract

In this paper, we consider the following coupled nonlinear Schrödinger system

$ \left\{\begin{array}{ll} -\Delta u+(1+\delta a(x))u = \mu_1 u^3+\beta uv^2 &\mbox{in }\mathbb{R}^3,\\ -\Delta v+(1+\delta b(x))v = \mu_2 v^3+\beta u^2v &\mbox{in }\mathbb{R}^3,\\ u\to 0,\quad v\to 0, &\mbox{as } |x|\to\infty \end{array}\right. $

where $ \mu_1>0 $, $ \mu_2>0 $, $ \beta\in\mathbb{R} $, $ \delta\in\mathbb{R} $, and $ a(x) $ and $ b(x) $ are two $ C^\alpha $ potentials with $ 0<\alpha<1 $, satisfying some slow decay assumptions, but do not need to fulfill any symmetry property. Using the Lyapunov–Schmidt reduction method and some variational techniques, we show that there exist $ 0<\delta_0<1 $ and $ 0<\beta_0<\min\{\mu_1,\mu_2\} $ such that the above system has infinitely many positive segregated solutions for any $ 0<\delta<\delta_0 $ and $ 0<\beta<\beta_0 $.

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Mathematics Subject Classification: Primary: 35J60, 35J75; Secondary: 35J80.

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