Concentrating ground state for linearly coupled Schrödinger systems involving critical exponent cases

Abstract

We study the following linearly coupled Schrödinger system:{−Δu+(μV1(x)+a)u=f(x)|u|p−2u+β(x)v,x∈RN,−Δv+(μV2(x)+b)v=g(x)|v|2⁎−2v+β(x)u,x∈RN, where N≥3,2<p≤2⁎, V1,V2∈C(RN,R+) are potential wells with bottoms Ωi=intVi−1(0), the parameters a>−λ1(Ω1),b>−λ1(Ω2) and λ1(Ωi) is the first eigenvalue of −Δ in H01(Ωi). Under some suitable assumptions on β(x) which relate to the potentials V1,V2 and the parameters a,b, the existence of positive ground states is obtained by variational method. Some interesting phenomena are that we relax the upper control condition of the coupling function β(x) and we do not need the weight function f(x) to be integrable or bounded in the subcritical case 2<p<2⁎. Moreover, from the concentration phenomenon of solutions, we obtain some results of the existence of positive ground states to a linearly coupled Schrödinger system in a bounded domain, which extend the recent results of Peng, et al. (2017), [25].