Ricerc@Sapienza

We study the ergodic problem for fully nonlinear operators which may be singular or degenerate when the gradient of solutions vanishes. We prove the convergence of both explosive solutions and solutions of Dirichlet problems for approximating equations. We further characterize the ergodic constant as the infimum of constants for which there exist bounded sub-solutions. As intermediate results of independent interest, we prove a priori Lipschitz estimates depending only on the norm of the zeroth order term, and a comparison principle for equations having no zero order terms.

Abstract. In this paper we study existence, nonexistence and classification of radial positive
solutions of some weighted fully nonlinear equations involving Pucci extremal operators. Our
results are entirely based on the analysis of the dynamics induced by an autonomous quadratic
system which is obtained after a suitable transformation. This method allows to treat both regular
and singular solutions in a unified way, without using energy arguments. In particular we recover