Gradient estimates for degenerate elliptic measure data problems with double phase

Abstract

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