Singular solutions of elliptic equations involving nonlinear gradient terms on perturbations of the ball

Abstract

In this article we obtain positive singular solutions of(1)−Δu=|∇u|p in Ω,u=0 on ∂Ω, where Ω is a small C2 perturbation of the unit ball in RN. For NN−1<p<2 we prove that if Ω is a sufficiently small C2 perturbation of the unit ball there exists a singular positive weak solution u of (1). In the case of p>2 we prove a similar result but now the positive weak solution u is contained in C0,p−2p−1(Ω‾) and yet is not in C0,p−2p−1+ε(Ω‾) for any ε>0.