Supercritical elliptic problems on nonradial domains via a nonsmooth variational approach

Abstract

In this paper we are interested in positive classical solutions of(1){−Δu=a(x)up−1 in Ω,u>0 in Ω,u=0 on ∂Ω, where Ω is a bounded annular domain (not necessarily an annulus) in RN(N≥3) 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 (1) has ⌊N2⌋ (the floor of N2) positive nonradial solutions when a(x)=1 and Ω 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.