Semiclassical states for coupled nonlinear Schrödinger equations with a critical frequency

Abstract

In this paper, we are concerned with the coupled nonlinear Schrödinger system \begin{align*} \begin{cases} -\varepsilon^{2}Δu+a(x)u=μ_{1}u^{3}+βv^{2}u \ \ \ \ \mbox{in}\ \mathbb{R}^{N},\\ -\varepsilon^{2}Δv+b(x)v=μ_{2}v^{3}+βu^{2}v \ \ \ \ \ \mbox{in}\ \mathbb{R}^{N}, \end{cases} \end{align*} where $1\le N\le 3$, ${\mu }_1,{\mu }_2,\beta >0$, $a(x)$ and $b(x)$ are nonnegative continuous potentials, and $\epsilon >0$ is a small parameter. We show the existence of positive ground state solutions for the system above and also establish the concentration behaviour as $\epsilon \to 0$, when $a(x)$ and $b(x)$ achieve 0 with a homogeneous behaviour or vanish in some nonempty open set with smooth boundary.

23 pages, typos corrected, to appear on Asymptotic Analysis