Singular solutions of some elliptic equations involving mixed absorption-reaction

We study properties of nonnegative functions satisfying (E)$\;-\Delta u+u^p-M|\nabla u|^q=0$ is a domain of $\mathbb{R}^N$ when $p>1$, $M>0$ and $1<q<p$. We concentrate our analysis on the solutions of (E) with an isolated singularity, or in an exterior domain, or in the whole space. The existence of such solutions and their behaviours depend strongly on the values of the exponents $p$ and $q$ and in particular according to the sign of $q-\frac{2p}{p+1}$, and when $q=\frac{2p}{p+1}$, also on the value of the parameter $M$ which becomes a key element. The description of the different behaviours is made possible by a sharp analysis of the radial solutions of (E).