Nehari-type ground state solutions for Schrödinger equations including critical exponent

Abstract

In this paper, we study the following system of nonlinear Schrödinger equations: { − △ u + a ( x ) u = | u | p − 2 u + λ ( x ) v , x ∈ R N , − △ v + b ( x ) v = | v | 2 ∗ − 2 v + λ ( x ) u , x ∈ R N , where N ≥ 3 , 2 < p < 2 ∗ and 2 ∗ = 2 N / ( N − 2 ) is the critical Sobolev exponent. Under assumptions that a ( x ) , b ( x ) , λ ( x ) ∈ C ( R N , R ) are all 1-periodic in each of x 1 , x 2 , … , x N and λ 2 ( x ) < a ( x ) b ( x ) , we prove that the above system has a Nehari-type ground state solution when 0 < a ( x ) < μ 0 for some μ 0 ∈ ( 0 , 1 ) .