Semiclassical states for coupled nonlinear Schrödinger equations with critical frequency

Abstract

In this paper, we are concerned with the coupled nonlinear Schrödinger system $$\begin{array}{l}\hfill \left\{\begin{array}{cc}-{\mathit{\epsilon }}^2\mathrm{\Delta }\mathit{u}+\mathit{a}\mathrm{(}\mathit{x}\mathrm{)}\mathit{u}={\mathit{\mu }}_1{\mathit{u}}^3+\mathit{\beta }{\mathit{v}}^2\mathit{u}\hfill & \text{in }{\mathbb{R}}^{\mathit{N}}\mathrm{,}\hfill \\ -{\mathit{\epsilon }}^2\mathrm{\Delta }\mathit{v}+\mathit{b}\mathrm{(}\mathit{x}\mathrm{)}\mathit{v}={\mathit{\mu }}_2{\mathit{v}}^3+\mathit{\beta }{\mathit{u}}^2\mathit{v}\hfill & \text{in }{\mathbb{R}}^{\mathit{N}}\mathrm{,}\hfill \end{array}\right.\end{array}$$ where $1⩽\mathit{N}⩽3$ , ${\mathit{\mu }}_1\mathrm{,}{\mathit{\mu }}_2\mathrm{,}\mathit{\beta }\mathrm{>}0$ , $\mathit{a}\mathrm{(}\mathit{x}\mathrm{)}$ and $\mathit{b}\mathrm{(}\mathit{x}\mathrm{)}$ are nonnegative continuous potentials, and $\mathit{\epsilon }\mathrm{>}0$ is a small parameter. We show the existence of positive ground state solutions for the system above and also establish the concentration behaviour as $\mathit{\epsilon }\to 0$ , when $\mathit{a}\mathrm{(}\mathit{x}\mathrm{)}$ and $\mathit{b}\mathrm{(}\mathit{x}\mathrm{)}$ achieve 0 with a homogeneous behaviour or vanish in some nonempty open set with smooth boundary.