Positive solutions of a nonlinear Schrödinger system with nonconstant potentials

Haidong  Liu; Zhaoli  Liu

doi:10.3934/dcds.2016.36.1431

Article Contents

2016, Volume 36, Issue 3: 1431-1464. Doi: 10.3934/dcds.2016.36.1431

This issue Previous Article Effective boundary conditions of the heat equation on a body coated by functionally graded material Next Article Young towers for product systems

Positive solutions of a nonlinear Schrödinger system with nonconstant potentials

Haidong Liu^1, and
Zhaoli Liu^2,

1.
College of Mathematics, Physics and Information Engineering, Jiaxing University, Zhejiang 314001
2.
School of Mathematical Sciences, Capital Normal University, Beijing 100048

Received: November 2014

Revised: June 2015

Published online: August 2015

Abstract / Introduction Related Papers Cited by

Abstract

Existence of a solution of the nonlinear Schrödinger system \begin{equation*} \left\{ \begin{aligned} & - \Delta u + V_1(x) u=\mu_1(x) u^3 + \beta(x) u v^2 \qquad\mbox{in}\ \mathbb{R}^N, \\ & - \Delta v + V_2(x) v=\beta(x) u^2 v + \mu_2(x) v^3 \qquad \mbox{in}\ \mathbb{R}^N, \\ & u>0,\ v>0,\quad u,\ v\in H^1(\mathbb{R}^N), \end{aligned} \right. \end{equation*} where $N=1,2,3$, and $V_j,\mu_j,\beta$ are continuous functions of $x\in\mathbb{R}^N$, is proved provided that either $V_j,\mu_j,\beta$ are invariant under the action of a finite subgroup of $O(N)$ or there is no such invariance assumption. In either case the result is obtained both for $\beta$ small and for $\beta$ large in terms of $V_j$ and $\mu_j$.

Keywords:

Nonlinear Schrödinger system,
variational methods.

Mathematics Subject Classification: Primary: 35J50; Secondary: 35A15.

Citation:

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