Principal eigenvalues and eigenfunctions for fully nonlinear equations in punctured balls

This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear 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(gamma) , u(gamma) ) of the equation F(D(2)u(gamma)) + lambda (gamma)u(gamma)/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) \ {0}/B(0, 1) and gamma > 0. We prove existence of radial solutions which are continuous on B (0 , 1) in the case gamma < 2, existence of unbounded solutions in the case gamma = 2 and a non existence result for gamma > 2. We also give, in the case of Pucci's operators, the explicit value of lambda(2) , which generalizes the Hardy-Sobolev constant for the Laplacian. (c) 2024 Published by Elsevier Masson SAS.