Gradient estimates for nonlinear elliptic equations with Orlicz growth and measure data

Abstract

We establish gradient estimates of solutions to a class of nonlinear elliptic equations with measure data under Orlicz-type growth conditions. The growth is governed by the structural condition \[ 0<i_a\le t g'(t)/g(t)\le s_a<1. \] We obtain two types of regularity results: pointwise Wolff potential estimates for the gradient of solutions in the singular regime $i_a \in \big(\frac{n-1}{2n-1},1\big)$, and Lipschitz regularity of the solutions in the regime $i_a \in (0,1)$. In the power-type case $g(t)=t^{p-1}$, our results recover the known gradient estimates for the singular $p$-Laplace equation.