Infinitely many solutions for the Hénon equation with critical exponent in non-convex domains

Abstract

We consider the Neumann problem for the Henon equation ⎧⎨ ⎩−Δu+ u = |x| u N+2 N−2 , u > 0, in Ω, ∂u ∂n = 0 on ∂Ω, (0.1) where Ω ⊂ R ,N 3 is a smooth and bounded domain, α > 0 and n denotes the outward unit normal vector of ∂Ω. We show that problem (0.1) has infinitely many positive solutions, whose energy can be made arbitrarily large in some (partially symmetric) non-convex domains Ω. This seems to be a new phenomenon for the Henon equation in bounded domains.