Gradient estimates for degenerate elliptic measure data problems with double phase

Abstract

We study nonlinear elliptic equations modeled on \[ -\mathrm{div}\,(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du) = μ, \] where $2\le p<q<\infty$, $a(\cdot )\ge 0$, and $\hat{I}¼$ is a signed Borel measure with finite total mass. We prove local Calderón--Zygmund type gradient estimates for SOLA (Solutions Obtained as Limits of Approximations) by finding new and natural assumptions on $p$, $q$ and $a(\cdot )$.