SUPER-LINEAR ELLIPTIC EQUATION FOR FULLY NONLINEAR OPERATORS WITHOUT GROWTH RESTRICTIONS FOR THE DATA

Abstract

In this paper we deal with existence and uniqueness of solution to the fully nonlinear equation −F(D2u) + |u|s−1u = f(x) in IRn, where s > 1 and f satisfies only local integrability conditions. This result is well known when, instead of the fully nonlinear elliptic operator F, the Laplacian or a divergence form operator is considered. Our existence results use the Alexandroff-Bakelman-Pucci inequality since we cannot use any variational formulation. For radially symmetric f, and in the particular case where F is a maximal Pucci operator, we can prove our results under less integrability assumptions, taking advantage of an appropriate variational formulation. We also obtain an existence result with boundary explosion in smooth domains.