In this paper we prove some Liouville theorems for nonnegative viscosity supersolutions of a class of fully nonlinear uniformly elliptic problems in R N .RÉSUMÉ. -Dans ce travail nous démontrons des théorèmes de Liouville pour des sur-solutions de viscosité positives de problèmes uniformement elliptique complètement non linéaires dans R N .

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. We prove a priori estimates and regularity results for some quasilinear degenerate elliptic equations arising in optimal stochastic control problems. Our main results show that strong coerciveness of gradient terms forces bounded viscosity subsolutions to be globally Hölder continuous, and solutions to be locally Lipschitz continuous. We also give an existence result for the associated Dirichlet problem.

We prove nonexistence results of Liouville type for nonnegative viscosity solutions of some equations involving the fully nonlinear degenerate elliptic operators Pk ±, defined respectively as the sum of the largest and the smallest k eigenvalues of the Hessian matrix. For the operator Pk + we obtain results analogous to those which hold for the Laplace operator in space dimension k. Whereas, owing to the stronger degeneracy of the operator Pk −, we get totally different results

In this note, we prove C 1,γ regularity for solutions of some fully nonlinear degenerate elliptic equations with "superlinear" and "subquadratic " Hamiltonian terms. As an application, we complete the results of [6] concerning the associated ergodic problem, proving, among other facts, the uniqueness, up to constants, of the ergodic function.