Entire solutions of fully nonlinear elliptic equations with a superlinear gradient term

TL;DR

This paper establishes the existence and uniqueness of entire viscosity solutions for fully nonlinear elliptic equations with superlinear gradient terms, extending classical results to more complex nonlinear operators.

Contribution

It introduces new methods to prove existence and uniqueness of solutions for fully nonlinear equations with superlinear gradient growth, without requiring global bounds.

Findings

Existence of entire viscosity solutions for a class of nonlinear elliptic equations.

Uniqueness results under convexity/concavity assumptions for specific gradient terms.

Handling superlinearity differently from linear cases to establish key properties.

Abstract

In this paper we consider second order fully nonlinear operators with an additive superlinear gradient term. Like in the pioneering paper of Brezis for the semilinear case, we obtain the existence of entire viscosity solutions, defined in all the space, without assuming global bounds. A uniqueness result is also obtained for special gradient terms, subject to a convexity/concavity type assumption where superlinearity is essential and has to be handled in a different way from the linear case.