Multiple solutions to a linearly coupled elliptic system with critical exponents

Abstract

We are concerned with the linearly coupled elliptic system with critical exponents{−Δu+a(x)u=|u|2⁎−2u+μ|u|p−2u+λv,x∈RN,−Δv+b(x)v=|v|2⁎−2v+ν|v|q−2v+λu,x∈RN,u,v∈H1(RN), where N≥3, 2<p,q<2⁎=2NN−2, μ,ν and λ are nonnegative constants, and a(x), b(x) are positive periodic functions. The existence of a positive ground state and a higher energy solution of this system is proved using variational methods. Additionally, the asymptotic behavior of these solutions as λ→0+ is obtained.