Normalized ground states for Schrödinger system with a coupled critical nonlinearity

Abstract

This paper deals with the existence of normalized solutions for the Schrödinger system with a coupled critical nonlinearity −Δu=λ1u+μ1|u|p−2u+βr1|u|r1−2u|v|r2,−Δv=λ2v+μ2|v|q−2v+βr2|u|r1|v|r2−2v,under the constraints ∫RN|u|2dx=a,∫RN|v|2dx=b,where a>0, b>0 are prescribed, N≥3, μ1,μ2,β>0, 2≤p,q<2+4N,r1,r2>1 and r1+r2=2∗≐2NN−2. The frequencies λ1 and λ2 appear as Lagrange multipliers. A positive ground state is obtained for small β>0.