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

In this article we obtain positive singular solutions of

(1)$?\mathrm{\Delta }u=|\mathrm{?}u{|}^p\text{ in }\mathrm{\Omega },u=0\text{ on }?\mathrm{\Omega },$

where Ω is a small $C^2$ perturbation of the unit ball in ${\mathbb{R}}^N$. For $\frac{N}{N?1}<p<2$ we prove that if Ω is a sufficiently small $C^2$ 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 $C^{0,\frac{p?2}{p?1}}(\stackrel{￣}{\mathrm{\Omega }})$ and yet is not in $C^{0,\frac{p?2}{p?1}+\epsilon }(\stackrel{￣}{\mathrm{\Omega }})$ for any $\epsilon >0$.