Singular solutions of a Hénon equation involving a nonlinear gradient term

Abstract

<p style='text-indent:20px;'>Here, we consider positive singular solutions of</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a small smooth perturbation of the unit ball in <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula> are parameters in a certain range. Using an explicit solution on <inline-formula><tex-math id="M5">\begin{document}$ B_1 $\end{document}</tex-math></inline-formula> and a linearization argument, we obtain positive singular solutions on perturbations of the unit ball.</p>