The H\'enon equation beyond the critical Sobolev exponent

Abstract

We consider the Dirichlet problem $-\Delta u=|x|^\alpha u^p$, $u>0$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega\subset R^N$, $N\ge 3$, is a bounded domain containing the origin and $\alpha>0$. We prove that a solution exists for every exponent $1<p<p_\alpha=\frac{N+2+2\alpha}{N-2}$, in particular in the whole Sobolev-supercritical range $\frac{N+2}{N-2}\le p<p_\alpha$, provided $\Omega$ is invariant under $O(N-1)\times O(1)$ 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.