Supercritical elliptic problems on nonradial domains via a nonsmooth variational approach

Abstract

In this paper we are interested in positive classical solutions of \begin{equation} \label{eqx} \left\{\begin{array}{ll} -\Delta u = a(x) u^{p-1} & \mbox{ in } \Omega, \\ u>0 & \mbox{ in } \Omega, \\ u= 0 & \mbox{ on } \pOm, \end {array}\right. \end{equation} where $\Omega$ is a bounded annular domain (not necessarily an annulus) in $\IR^N$ $(N \ge3)$ and $ a(x)$ is a nonnegative continuous function. We show the existence of a classical positive solution for a range of supercritical values of $p$ when the problem enjoys certain mild symmetry and monotonicity conditions. As a consequence of our results, we shall show that (\ref{eqx}) has $\Bigl\lfloor\frac{N}{2} \Bigr\rfloor$ (the floor of $\frac{N}{2}$) positive nonradial solutions when $ a(x)=1$ and $\Omega$ is an annulus with certain assumptions on the radii. We also obtain the existence of positive solutions in the case of toroidal domains. Our approach is based on a new variational principle that allows one to deal with supercritical problems variationally by limiting the corresponding functional on a proper convex subset instead of the whole space at the expense of a mild invariance property.