Supercritical elliptic problem with nonautonomous nonlinearities

In this paper we study the problemequation(0.1){−Δu=|x|αupα−ϵin Ω,u>0in Ω,u=0on ∂Ω, where pα=N+2+2αN−2, Ω   is a smooth bounded domain of RNRN, 0∈Ω0∈Ω, N⩾3N⩾3 and α∈(0,1]α∈(0,1]. We show that, for ϵ   small enough, there exists at least one solution concentrating at x=0x=0.