On radial solutions for a Fully Non Linear degenerate or singular PDE

In this paper we consider equations $-| \nabla u |^\alpha F ( D^2 u) = |u|^{p-1} u$ in an annulus. $F$ is Fully Nonlinear Elliptic, $\alpha$ is some real $> -1$ and $p > 1+ \alpha$. The solutions are intended in the sense of the definition given by Birindelli and Demengel \cite{BD1}. We prove the existence of non trivial positive/negative radial solutions.