# Pointwise gradient bounds for entire solutions of elliptic equations with non-standard growth conditions and general nonlinearities

## TL;DR

This paper establishes pointwise gradient bounds for solutions to very general elliptic PDEs with non-standard growth, including singular, degenerate, and nonlinear cases, in the entire Euclidean space.

## Contribution

It provides a unified framework for gradient bounds applicable to a broad class of elliptic equations with complex nonlinearities and growth conditions.

## Key findings

- Derived pointwise gradient bounds for solutions
- Applicable to singular and degenerate nonlinear cases
- Handles general nonlinear source terms

## Abstract

We give pointwise gradient bounds for solutions of (possibly non-uniformly) elliptic partial differential equations in the entire Euclidean space. The operator taken into account is very general and comprises also the singular and degenerate nonlinear case with non-standard growth conditions. The sourcing term is also allowed to have a very general form, depending on the space variables, on the solution itself, on its gradient, and possibly on higher order derivatives if additional structural conditions are satisfied.