The supercritical Lane-Emden-Fowler equation in exterior domains

Abstract

We consider the exterior problem

Delta u + u(p) = 0, u > 0 in IRN\(D) over bar,

u = 0 on partial derivative D, lim(vertical bar x vertical bar ->+infinity) u(x) = 0

where D is a bounded, smooth domain in IRN, for supercritical powers p > 1. We prove that if N > 4 and p > N-3/N-3, then this problem admits infinitely many solutions. If D is symmetric with respect to N axes, this result holds whenever N >= 3 and p > N+2/N-2.