Gradient estimates of very weak solutions to general quasilinear elliptic equations

TL;DR

This paper derives gradient estimates for very weak solutions to general quasilinear elliptic equations with nonstandard growth, extending known results without regularity assumptions and establishing higher integrability and existence results.

Contribution

It provides new gradient estimates and existence results for very weak solutions to quasilinear elliptic equations with nonstandard growth, without regularity constraints on the nonlinearity.

Findings

Gradient estimate for very weak solutions established

Higher integrability of the gradient proven

Existence of very weak solutions demonstrated

Abstract

We establish a gradient estimate for a very weak solution to a quasilinear elliptic equation with a nonstandard growth condition, which is a natural generalization of the $p$-Laplace equation. We investigate the maximum extent for the gradient estimate to hold without imposing any regularity assumption on the nonlinearity other than basic structure assumptions. Our results also include a higher integrability result of the gradient and the existence for the very weak solutions to such nonlinear problems.