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

Abstract

We consider the Neumann problem for the Hénon equation

 −Δu+u=|x|^2αu^{(N+2)/(N−2)}, u>0, in Ω,

 ∂u/∂n=0   on ∂Ω,  (0.1)

where Ω⊂R^N,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 Hénon equation in bounded domains.