Nonlinear elliptic equations with high order singularities

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.Comment: Revise

Sharp regularity for general Poisson equations with borderline sources

This article concerns optimal estimates for non-homogeneous degenerate elliptic equation with source functions in borderline spaces of integrability. We deliver sharp H\"older continuity estimates for solutions to $p$-degenerate elliptic equations in rough media with sources in the weak Lebesgue space $L_\text{weak}^{\frac{n}{p} + \epsilon}$. For the borderline case, $f \in L_\text{weak}^{\frac{n}{p}}$, solutions may not be bounded; nevertheless we show that solutions have bounded mean oscillation, in particular John-Nirenberg's exponential integrability estimates can be employed. All the results presented in this paper are optimal. Our approach is based on powerful Caffarelli-type compactness methods and it can be employed in a number order situations.Comment: Review from previous version. Accepted for Publication: Journal de Math\'ematiques Pures et Appliqu\'ee