The Hénon equation beyond the critical Sobolev exponent

We consider the Dirichlet problem -Δ u=|x|^α uᵖ, $u>0$ in Ω, $u=0$ on ∂Ω, where Ω⊂ RN, N≥ 3, is a bounded domain containing the origin and α>0. We prove that a solution exists for every exponent 1<p<p_α=N+2+2α/N-2, in particular in the whole Sobolev-supercritical range N+2/N-2≤ p<p_α, provided Ω is invariant under O(N-1)× O(1) and satisfies an angular monotonicity condition. This class contains the balls, the oblate ellipsoids of revolution, lens-shaped domains, and non-convex, non-smooth and non star-shaped domains. The solution is obtained by minimizing the Rayleigh quotient on a suitable closed convex cone.