Principal eigenvalues for Fully Non Linear singular or degenerate operators in punctured balls

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 ((lambda) over bar(gamma), u(gamma)) of the equation |del u|(alpha) F(D(2)u(gamma)) + (lambda) over bar(gamma) u(gamma)(1+alpha)/r(gamma )= 0 in B(0, 1) \ {0}, u(gamma )= 0 on partial derivative 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 (B(0, 1)) over bar in the case gamma < 2 + alpha, and a non existence result for gamma < 2 + alpha. We also give the explicit value of (lambda) over bar(2+alpha) in the case of the Pucci's operators, which generalizes the Hardy-Sobolev constant for the Laplacian, and the previous results of Birindelli et al. [1].