Existence of standing wave solutions for coupled quasilinear Schr?dinger systems with critical exponents in R

摘要

Abstract: This paper is concerned with the following quasilinear Schr?dinger system in RNRN:{?ε2Δu+V1(x)u?ε2Δ(u2)u=K1(x)|u|22??2u+h1(x,u,v)u,?ε2Δv+V2(x)v?ε2Δ(v2)v=K2(x)|v|22??2v+h2(x,u,v)v,{?ε2Δu+V1(x)u?ε2Δ(u2)u=K1(x)|u|22??2u+h1(x,u,v)u,?ε2Δv+V2(x)v?ε2Δ(v2)v=K2(x)|v|22??2v+h2(x,u,v)v,where N≥3N≥3, Vi(x)Vi(x) is a nonnegative potential, Ki(x)Ki(x) is a bounded positive function, i=1,2.i=1,2. h1(x,u,v)uh1(x,u,v)u and h2(x,u,v)vh2(x,u,v)v are superlinear but subcritical functions. Under some proper conditions, minimax methods are employed to establish the existence of standing wave solutions for this system provided that εε is small enough, more precisely, for any m∈Nm∈N, it has mm pairs of solutions if εε is small enough. And these solutions (uε,vε)→(0,0)(uε,vε)→(0,0) in some Sobolev space as ε→0ε→0. Moreover, we establish the existence of positive solutions when ε=1ε=1. The system studied here can model some interaction phenomena in plasma physics.