Affiliations: [a] Department of Mathematics, Federal University of Paraíba, João Pessoa, PB, Brazil. E-mail: jmbo@pq.cnpq.br | [b] Institute of Mathematics and Statistics, Federal University of Goiás, Goiás, GO, Brazil. E-mail: joserre@gmail.com
Abstract: We study the following class of linearly coupled Schrödinger elliptic systems
−Δu+V1(x)u=μ|u|p−2u+λ(x)v,x∈RN,−Δv+V2(x)v=|v|q−2v+λ(x)u,x∈RN,
where N⩾3, 2<p⩽q⩽2∗=2N/(N−2) and μ⩾0. We consider nonnegative potentials periodic or asymptotically periodic which are related with the coupling term λ(x) by the assumption |λ(x)|⩽δV1(x)V2(x), for some 0<δ<1. We deal with three cases: Firstly, we study the subcritical case, 2<p⩽q<2∗, and we prove the existence of positive ground state for all parameter μ⩾0. Secondly, we consider the critical case, 2<p<q=2∗, and we prove that there exists μ0>0 such that the coupled system possesses positive ground state solution for all μ⩾μ0. In these cases, we use a minimization method based on Nehari manifold. Finally, we consider the case p=q=2∗, and we prove that the coupled system has no positive solutions. For that matter, we use a Pohozaev identity type.
Keywords: Coupled systems, nonlinear Schrödinger equations, lack of compactness, ground states
