Super-linear elliptic equation for the Pucci operator without growth restrictions for the data

TL;DR

This paper establishes existence and uniqueness of solutions for super-linear elliptic equations involving the Pucci operator under minimal data restrictions, extending known results beyond classical operators like Laplacian.

Contribution

It introduces new existence and uniqueness results for Pucci operator equations without growth restrictions on data, using ABP inequality and variational methods for symmetric cases.

Findings

Existence and uniqueness for super-linear Pucci equations with minimal data assumptions.

Results extend classical PDE theory to fully nonlinear operators.

Additional solutions with boundary explosion in smooth domains.

Abstract

In this paper we deal with existence and uniqueness of solution to super-linear problems for the Pucci operator: $$ -\M^+(D^2u)+|u|^{s-1}u=f(x) \quad {in} \RR^n, $$ where $s>1$ and $f$ satisfies only local integrability conditions. This result is well known when, instead of the Pucci operator, 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 $f$ we can prove our results under less local integrability assumptions, taking advantage of an appropriate variational formulation. We also obtain an existence result with boundary explosion in smooth domains.