H\"older regularity and Liouville properties for nonlinear elliptic inequalities with power-growth gradient terms

TL;DR

This paper establishes sharp regularity and Liouville properties for nonlinear elliptic inequalities with gradient terms of power growth, using integral methods that do not rely on maximum principles.

Contribution

It provides new proofs of regularity and Liouville theorems for a broad class of nonlinear elliptic inequalities, extending results to sub-Riemannian and Riemannian manifold settings.

Findings

Proves sharp local H"older regularity for solutions.

Establishes Liouville properties without one-sided bounds.

Extends results to geometric contexts like sub-Riemannian and Riemannian manifolds.

Abstract

This note studies local integral gradient bounds for distributional solutions of a large class of partial differential inequalities with diffusion in divergence form and power-like first-order terms. The applications of these estimates are two-fold. First, we show the (sharp) global H\"older regularity of distributional semi-solutions to this class of diffusive PDEs with first-order terms having supernatural growth and right-hand side in a suitable Morrey class posed on a bounded and regular open set $\Omega$. Second, we provide a new proof of entire Liouville properties for inequalities with superlinear first-order terms without assuming any one-side bound on the solution for the corresponding homogeneous partial differential inequalities. We also discuss some extensions of the previous properties to problems arising in sub-Riemannian geometry and also to partial differential inequalities posed on noncompact complete Riemannian manifolds under appropriate area-growth conditions of the geodesic spheres, providing new results in both these directions. The methods rely on integral arguments and do not exploit maximum and comparison principles.