Ground states of nonlinear Schrödinger systems with mixed couplings: the critical case

Abstract

In this paper, we consider the following k-coupled nonlinear Schrödinger systems in the critical case: Here, is a smooth bounded domain, and for every , where is the first eigenvalue of with the Dirichlet boundary condition. Note that the nonlinearity and the coupling terms are both critical in dimension 4 (i.e. when ). We call the couplings are attractive if , while repulsive stands for . Under the assumption that all the couplings are purely attractive and large enough, we first show that this critical system has a fully nontrivial ground state solution, that is, a solution has all components nontrivial, under conditions providing additional , while ground state solution may be semitrivial ( has null components) without the above additional conditions. When the systems admit mixed couplings, i.e. there exist and such that and , we establish the existence of least energy positive solutions. The purpose of this paper is to solve some remaining open questions on Tavares and You (Calc. Var. Partial Differential Equations 59:26, 2020).