Abstract :
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.
