Semiclassical states for coupled nonlinear Schr\"{o}dinger equations with a critical frequency

TL;DR

This paper proves the existence and concentration behavior of positive ground state solutions for a coupled nonlinear Schrödinger system with critical frequency, as the small parameter approaches zero, under various potential conditions.

Contribution

It establishes new existence results and analyzes the concentration phenomena of solutions for coupled Schrödinger equations with critical frequency and variable potentials.

Findings

Existence of positive ground state solutions.

Concentration behavior as epsilon approaches zero.

Solutions localize near potential minima or vanishing regions.

Abstract

In this paper, we are concerned with the coupled nonlinear Schr\"{o}dinger system \begin{align*} \begin{cases} -\varepsilon^{2}\Delta u+a(x)u=\mu_{1}u^{3}+\beta v^{2}u \ \ \ \ \mbox{in}\ \mathbb{R}^{N},\\ -\varepsilon^{2}\Delta v+b(x)v=\mu_{2}v^{3}+\beta u^{2}v \ \ \ \ \ \mbox{in}\ \mathbb{R}^{N}, \end{cases} \end{align*} where $1\leq N\leq3$, $\mu_{1},\mu_{2},\beta>0$, $a(x)$ and $b(x)$ are nonnegative continuous potentials, and $\varepsilon>0$ is a small parameter. We show the existence of positive ground state solutions for the system above and also establish the concentration behaviour as $\varepsilon\rightarrow0$, when $a(x)$ and $b(x)$ achieve 0 with a homogeneous behaviour or vanish in some nonempty open set with smooth boundary.