Existence and nonexistence of viscosity solutions for a class of degenerate/singular eigenvalue type equations

TL;DR

This paper thoroughly classifies when viscosity solutions exist or do not exist for a class of degenerate or singular eigenvalue equations, providing global estimates and employing advanced methods like Perron and comparison principles.

Contribution

It offers a complete classification of existence and nonexistence of viscosity solutions for eigenvalue equations with boundary degeneracy or singularity, including all cases related to the distance function exponent.

Findings

Complete classification of solution existence based on boundary behavior.

Derivation of global estimates for solutions when they exist.

Development of methods adapting Perron and comparison principles for degenerate/singular equations.

Abstract

This paper is devoted to a complete classification on the existence and nonexistence results of viscosity solutions to the general Dirichlet problem for a class of eigenvalue type equations. With the distance function included in the right-hand side, this type of equations can be degenerate and (or) singular near the boundary of uniformly convex domains. One highlight is that all cases related to the exponent of the distance function are investigated. Moreover, when viscosity solutions exist, we derive a series of global estimates based on the distance function. The key ingredients of this paper include adaptions of the Perron method and comparison principle as well as constructions of suitable classical sub-solutions and super-solutions.