Asymptotic behavior of Solutions for Hénon systems with nearly critical exponent

We consider in this paper the problemequation(0.1){−Δu=|x|αvp,−Δv=|x|βuqε,x∈Ω,u>0,v>0,x∈Ω,u=v=0,x∈∂Ω, where Ω   is the unit ball in RNRN centered at the origin, 0⩽α0β>0, N⩾8N⩾8, p>1p>1, qε>1qε>1. Suppose qε→q>1qε→q>1 as ε→0+ε→0+ and qε,qqε,q satisfy respectivelyNp+1+Nqε+1>N−2,Np+1+Nq+1=N−2, we investigate the asymptotic behavior of the ground state solutions (uε,vε)(uε,vε) of (0.1) as ε→0+ε→0+. We show that the ground state solutions concentrate at a point, which is located at the boundary. In addition, the ground state solution is non-radial provided that ε>0ε>0 is small.