Ground states for a system of Schrödinger equations with critical exponent

Abstract

We study the following system of nonlinear Schrödinger equations:{−Δu+μu=|u|p−1u+λv,x∈RN,−Δv+νv=|v|2⁎−2v+λu,x∈RN, where N⩾3, 2⁎=2NN−2, 1<p<2⁎−1 and μ,ν,λ are positive parameters satisfying 0<λ<μν. We show that, there is some critical value μ0∈(0,1), such that this system has a positive ground state solution if 0<μ⩽μ0. In the case μ>μ0, there exists λμ,ν∈[(μ−μ0)ν,μν) such that, this system has no ground state solutions if λ<λμ,ν; while this system has a positive ground state solution if λ>λμ,ν. In particular, if p=2⁎−1, the system has no nontrivial solutions. Some further properties of the ground state solutions are also studied. This seems to be the first result for such a critical Schrödinger system.