Semiclassical states for coupled nonlinear Schrödinger equations with critical frequency

Abstract

In this paper, we are concerned with the coupled nonlinear Schrödinger system − ε 2 Δ u + a ( x ) u = μ 1 u 3 + β v 2 u in  R N , − ε 2 Δ v + b ( x ) v = μ 2 v 3 + β u 2 v in  R N , where 1 ⩽ N ⩽ 3, μ 1 , μ 2 , β > 0, a ( x ) and b ( x ) are nonnegative continuous potentials, and ε > 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 ε → 0, when a ( x ) and b ( x ) achieve 0 with a homogeneous behaviour or vanish in some nonempty open set with smooth boundary.