Comparison principle for unbounded viscosity solutions of degenerate elliptic PDEs with gradient superlinear terms

TL;DR

This paper establishes comparison principles for unbounded viscosity solutions of degenerate elliptic PDEs with superlinear gradient terms, expanding understanding of solution behavior under polynomial growth conditions.

Contribution

It introduces new comparison principles applicable to unbounded solutions of degenerate elliptic PDEs with superlinear gradient growth, including nonconvex cases and applications to PDE systems.

Findings

Comparison principles for unbounded solutions established

Results applicable to convex and some nonconvex PDEs

Applications demonstrated in monotone PDE systems

Abstract

We are concerned with fully nonlinear possibly degenerate elliptic partial differential equations (PDEs) with superlinear terms with respect to $Du$. We prove several comparison principles among viscosity solutions which may be unbounded under some polynomial-type growth conditions. Our main result applies to PDEs with convex superlinear terms but we also obtain some results in nonconvex cases. Applications to monotone systems of PDEs are given.