Gradient estimates for degenerate elliptic measure data problems with double phase

We study nonlinear elliptic equations modeled on $$-\mathrm{d}\mathrm{i}\mathrm{v}(\mathrm{\mid }Du{\mathrm{\mid }}^{p-2}Du+a(x)\mathrm{\mid }Du{\mathrm{\mid }}^{q-2}Du)=\mu ,$$ where $2\le p<q<\mathrm{\infty }$, $a(\cdot )\ge 0$, and $\mu$ 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 $a(\cdot )$.