  This paper is devoted to the proof of the existence of the principal
eigenvalue and related eigenfunctions for fully nonlinear degenerate or
singular uniformly elliptic equations posed in a punctured ball, in presence of
a singular potential. More precisely, we analyze existence, uniqueness and
regularity of solutions $( \bar\lambda_\gamma, u_\gamma)$ of the equation $$|
\nabla u |^\alpha F( D^2 u_\gamma)+ \bar \lambda_\gamma {u_\gamma^{1+\alpha}
\over r^\gamma} = 0\ {\rm in} \ B(0,1)\setminus \{0\}, \ u_\gamma = 0 \ {\rm
on} \ \partial B(0,1)$$
  where $u_\gamma>0$ in $B(0,1)$, $\alpha >-1$ and $\gamma >0$. We prove
existence of radial solutions which are continuous on $\overline{ B(0,1)}$ in
the case $\gamma <2+\alpha$, existence of unbounded solutions which do ot
satisfy the boundary condition in the case $\gamma = 2+\alpha $ and a non
existence result for $\gamma >2+\alpha$.
  We also give the explicit value of $\bar \lambda_{2+\alpha} $ in the case of
Pucci's operators, which generalizes the Hardy--Sobolev constant for the
Laplacian, and the previous results of Birindelli, Demengel and Leoni
