Standing waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^N$

Abstract

We study existence and concentration of positive solutions for systems of the form $-\epsilon^2\Delta u + V(x)u = \nabla F(u)$ in $\mathbb{R}^N$ where $\epsilon \gt 0$ is a small parameter, $F : [0,\infty)^k \rightarrow \mathbb{R}$ is a $C_{loc}^{1,\alpha}$ function, $N \geq 3, k \geq 1$, and the potential $V$ has a positive infimum and a well.