Abstract
In this article we obtain positive singular solutions of(1){−Δpu=|∇u|qinΩ,u=0on∂Ω, where Ω is a small C2 perturbation of the unit ball in RN. For (p−1)NN−1<q<p<N 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). For other ranges of p and q we prove the existence of Hölder continuous positive solution (with optimal regularity) on a C2 perturbation of the unit ball.
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