Local behaviour of the solutions of the Chipot-Weissler equation

We study the local properties of positive solutions of the equation $-\mathrm{\Delta }u=u^p-m\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^q$ in a punctured domain $\mathrm{\Omega }\setminus \{0\}$ 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 ${\mathbb{R}}^N$.