Nonlinear elliptic equations with high order singularities

TL;DR

This paper investigates complex nonlinear elliptic equations with interior singularities, establishing universal continuity and sharp regularity estimates for solutions without boundary conditions.

Contribution

It introduces new regularity results for high order singular degenerate elliptic equations with interior singularities, independent of boundary data.

Findings

Positive solutions have a universal modulus of continuity.

Established sharp regularity estimates for limiting solutions.

Solutions exhibit regularity regardless of their infimum value.

Abstract

We study non-variational degenerate elliptic equations with high order singular structures. No boundary data are imposed and singularities occur along an {\it a priori} unknown interior region. We prove that positive solutions have a universal modulus of continuity that does not depend on their infimum value. We further obtain sharp, quantitative regularity estimates for non-negative limiting solutions.