Asymptotic behavior of Solutions for Hénon systems with nearly critical exponent

We consider in this paper the problem ⎧⎨⎩ − u = |x|α v p, − v = |x|βuqε , x ∈ Ω, u > 0, v > 0, x ∈ Ω, u = v = 0, x ∈ ∂Ω, (0.1) where Ω is the unit ball in RN centered at the origin, 0 α < pN, β >0, N 8, p > 1, qε > 1. Suppose qε →q >1 as ε→0+ and qε, q satisfy respectively N p +1 + N qε +1 > N −2, N p +1 + N q +1 = N −2, we investigate the asymptotic behavior of the ground state solutions (uε, vε) of (0.1) as 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 > 0 is small.