Harnack inequality for anisotropic fully nonlinear equations with nonstandard growth

TL;DR

This paper proves Harnack inequalities for a class of degenerate anisotropic fully nonlinear elliptic equations with nonstandard growth, using a novel barrier function and intrinsic geometry techniques.

Contribution

It introduces a new approach with a tailored barrier function and extends Harnack inequalities to anisotropic equations with nonstandard growth conditions.

Findings

Established Harnack inequalities for anisotropic degenerate equations.

Developed a new barrier function for doubling property.

Extended intrinsic Harnack inequalities to nonstandard growth operators.

Abstract

We establish Harnack inequalities for viscosity solutions of a class of degenerate fully nonlinear anisotropic elliptic equations exhibiting non-standard growth conditions. A primary example of such operators is the degenerate anisotropic $(p_i)$-Laplacian. Our approach relies on the sliding paraboloid method, adapted with suitably chosen anisotropic functions to derive the basic measure estimates. A central contribution of this work is the development of a doubling property, achieved through the explicit construction of a novel barrier function. By combining these tools with the intrinsic geometry techniques introduced in [DGV08, VV25], we prove the intrinsic Harnack inequality for this class of operators under appropriate conditions on the exponents $(p_i)$.