Singular solutions of a Hénon equation involving a nonlinear gradient term

Here, we consider positive singular solutions of\begin{document}$ \begin{equation*} \left\{ \begin{array}{lcc} -\Delta u = |x|^\alpha |\nabla u|^p & \text{in}& \Omega \backslash\{0\},\\ u = 0&\text{on}& \partial \Omega, \end{array} \right. \end{equation*} $\end{document}where \begin{document}$ \Omega $\end{document} is a small smooth perturbation of the unit ball in \begin{document}$ \mathbb{R}^N $\end{document} and \begin{document}$ \alpha $\end{document} and \begin{document}$ p $\end{document} are parameters in a certain range. Using an explicit solution on \begin{document}$ B_1 $\end{document} and a linearization argument, we obtain positive singular solutions on perturbations of the unit ball.