New type of solutions to a slightly subcritical Hénon type problem on general domains

Abstract

We consider the following slightly subcritical problem(℘ε){−Δu=β(x)|u|p−1−εuin Ω,u=0on ∂Ω, where Ω is a smooth bounded domain in Rn, 3≤n≤6, p:=n+2n−2 is the Sobolev critical exponent, ε is a small positive parameter and β∈C2(Ω‾) is a positive function. We assume that there exists a nondegenerate critical point ξ⁎∈∂Ω of the restriction of β to the boundary ∂Ω such that∇(β(ξ⁎)−2p−1)⋅η(ξ⁎)>0, where η denotes the inner normal unit vector on ∂Ω. Given any integer k≥1, we show that for ε>0 small enough problem (℘ε) has a positive solution, which is a sum of k bubbles which accumulate at ξ⁎ as ε tends to zero. We also prove the existence of a sign changing solution whose shape resembles a sum of a positive bubble and a negative bubble near the point ξ⁎.