  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).
