On the viscosity solutions to some nonlinear elliptic equations

TL;DR

This paper studies viscosity solutions of certain nonlinear degenerate elliptic equations, establishing comparison principles, bounds, and eigenvalue existence results to advance understanding of their properties.

Contribution

It introduces new comparison principles and eigenvalue existence results for viscosity solutions of nonlinear degenerate elliptic equations.

Findings

Established comparison principles for solutions.

Proved a priori supremum bounds.

Demonstrated existence of first eigenvalue and positive eigenfunction.

Abstract

We consider viscosity solutions of a class of nonlinear degenerate elliptic equations on bounded domains. We prove comparison principles and a priori supremum bounds for the solutions. We also address the eigenvalue problem and, in many instances, show the existence of a first eigenvalue and a first positive eigenfunction.