Linear perturbations for the critical Hénon problem

In this paper, we study the problem   −∆u = |x| α u p α + |x| β u in Ω u> 0 in Ω  u =0 on ∂Ω, (0.1) , Ω is a smooth bounded domain of R N with 0 ∈ Ω where p α = N +2+2α N −2 and N ≥ 4. We show that, for α ≥ 0 and 0 ≤ β ≤ N − 4, there exists one solution concentrating at x = 0 as → 0. Moreover, we prove that, if Ω is a ball, there exist no radial solution if α = β > N − 4