# Regularity of solutions to a class of variable-exponent fully nonlinear elliptic equations

## TL;DR

This paper establishes regularity results for viscosity solutions to a broad class of variable-exponent fully nonlinear elliptic equations, including degenerate and singular cases, with estimates independent of the continuity of the exponent.

## Contribution

It proves local $C^{1,
u}$ regularity for solutions to variable-exponent elliptic equations under general conditions, extending the theory to discontinuous and degenerate cases.

## Key findings

- Viscosity solutions are locally $C^{1,
u}$ regular.
- Regularity estimates do not depend on the modulus of continuity of the exponent.
- Results apply to degenerate and blow-up ellipticity scenarios.

## Abstract

In the present paper, we propose the investigation of variable-exponent, degenerate/singular elliptic equations in non-divergence form. This current endeavor parallels the by now well established theory of functionals satisfying nonstandard growth condition, which in particular encompasses problems ruled by the $p(x)$-laplacian operator. Under rather general conditions, we prove viscosity solutions to variable exponent fully nonlinear elliptic equations are locally of class $C^{1,\kappa}$ for a universal constant $0< \kappa < 1$. A key feature of our estimates is that they do not depend on the modulus of continuity of exponent coefficients, and thus may be employed to investigate a variety of problems whose ellipticity degenerates and/or blows-up in a discontinuous fashion.