Electron. J. Diff. Equ., Vol. 2014 (2014), No. 251, pp. 1-11.

Semiclassical solutions for linearly coupled Schrodinger equations

Sitong Chen, Xianhua Tang

Abstract:
We consider the system of coupled nonlinear Schrodinger equations
$$\displaylines{ -\varepsilon^2\Delta u+a(x) u=H_{u}(x, u, v)+\mu(x) v, \quad x\in \mathbb{R}^N,\cr -\varepsilon^2\Delta v+b(x) v=H_{v}(x, u, v)+\mu(x) u, \quad x\in \mathbb{R}^N,\cr u,v\in H^1(\mathbb{R}^N), }$$
where $N\geq 3$, $a, b, \mu \in C(\mathbb{R}^N)$ and $H_{u}, H_{v}\in C(\mathbb{R}^N\times \mathbb{R}^2, \mathbb{R})$. Under conditions that $a_0=\inf a=0$ or $b_0=\inf b=0$ and $|\mu(x)|^2\le \theta a(x)b(x)$ with $\theta\in(0, 1)$ and some mild assumptions on $H$, we show that the system has at least one nontrivial solution provided that $0<\varepsilon\le \varepsilon_0$, where the bound $\varepsilon_0$ is formulated in terms of N, a, b and H.

Submitted October 23, 2014. Published December 1, 2014.
Math Subject Classifications: 35J20, 58E50.
Key Words: Nonlinear Schrodinger equation; semiclassical solution; coupled system;

Show me the PDF file (252 KB), TEX file, and other files for this article.

Sitong Chen
School of Mathematics and Statistics
Central South University
Changsha, Hunan 410083, China
email:sitongchen2041@hotmail.com
Xianhua Tang
School of Mathematics and Statistics
Central South University
Changsha, Hunan 410083, China
email: tangxh@mail.csu.edu.cn

Return to the EJDE web page