In this work we consider the existence of positive singular solutions
$ \begin{equation} \left\{ \begin{array}{lcl} \hfill -\Delta u_1 & = & \lambda_1 | \nabla u_2|^p \qquad \mbox{ in } \Omega, \\ \hfill -\Delta u_2 & = & \lambda_2 | \nabla u_1|^q \qquad \mbox{ in } \Omega, \\ \hfill u_1 = u_2 & = & 0 \hfill \mbox{ on } \partial \Omega, \end{array}\right.\;\;\;\;\;\;\;(1) \end{equation} $
where $ \Omega $ is small $ C^2 $ perturbation of the unit ball $ B_1 $ in $ \mathbb{R}^N $ and $ \lambda_i $ are positive constants. Under suitable conditions on $ p $ and $ q $ we prove the existence of positive singular solutions of (1). We also examine the case where one or both of $ u_1,u_2 $ are Hölder continuous.
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