Supercritical elliptic problems on nonradial domains via a nonsmooth variational approach

TL;DR

This paper establishes the existence of positive solutions for supercritical elliptic problems on nonradial domains using a novel nonsmooth variational approach, revealing multiple nonradial solutions under certain symmetry and monotonicity conditions.

Contribution

Introduces a new variational method for supercritical elliptic problems on nonradial domains, enabling the existence of multiple positive solutions.

Findings

Existence of positive solutions for supercritical exponents p.

Multiple nonradial solutions on annular and toroidal domains.

A new variational principle handling supercritical problems.

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.