Nonlinear elliptic problems above criticality

Abstract

We consider the elliptic problem Δu + up = 0, u > 0 in an exterior domain, Ω=RN∖D under zero Dirichlet and vanishing conditions, where D is smooth and bounded, and p is supercritical, namely p>N+2N−2 . We prove that this problem has infinitely many solutions with slow decay O(|x|−2p−1) at infinity. In addition, a fast decay solution exists if p is close enough to the critical exponent. If p differs from certain sequence of resonant values which tends to infinity, then the Dirichlet problem is also solvabe in a bounded domain Ω with a sufficiently small spherical hole.