Local behaviour of the solutions of the Chipot-Weissler equation

Abstract: We study the local properties of positive solutions of the equation $-\Delta u=u^p-m|\nabla u|^q$ in a punctured domain $\Omega\setminus{0}$ of $\mathbb{R}^N$ or in a exterior domain $\mathbb{R}^N\setminus B_{r_0}$ in the range $\min{p,q}>1$ and $m>0$. We prove a series of a priori estimates depending $p$ and $q$, and of the sign of $q-\frac {2p}{p+1}$ and $q-p$. Using various techniques we obtain removability results for singular sets and we give a precise description of behaviour of solutions near an isolated singularity or at infinity in $\mathbb{R}^N$.