Singular solutions of a Hénon equation involving a nonlinear gradient term (Q2070067)

This article investigates the existence of positive singular solutions to \(-\Delta u=|x|^\alpha |\nabla u|^p\) in \(\Omega\), subject to the boundary condition \(u=0\) on \(\partial \Omega\). Here \(\Omega\subset \mathbb{R}^N\) is a small smooth perturbation of the unit ball in \(\mathbb{R}^N\), \(\alpha>-1\) and either \((N+\alpha)/(N-1)<p<2+\alpha\) or \(p>2+\alpha\). The main result established the existence of a positive singular solution whose behaviour around the origin is given by a specific radial function. The approach combines linear theory and a fixed point argument on the deformation of the ball.