Ground states for a system of Schrödinger equations with critical exponent

We study the following system of nonlinear Schrödinger equations: − u +μu = |u|p−1u+λv, x ∈ RN, − v +νv = |v|2∗−2v +λu, x ∈ RN, where N 3, 2∗ = 2N N−2, 1μ0, 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. © 2012 Elsevier Inc. All rights reserved.