Global Lorentz gradient estimates for quasilinear equations with measure data for the strongly singular case: $1<p\le \frac{3n-2}{2n-1}$

In this paper, we study the global regularity estimates in Lorentz spaces for gradients of solutions to quasilinear elliptic equations with measure data of the form \begin{eqnarray*} \left{ \begin{array}{rcl} -{\rm div}($\mathcal{A}$(x, \nabla u))&=& \mu \quad $\text{in}$ ~\Omega, u&=&0 \quad $\text{on}$~ \partial \Omega, \end{array}\right. \end{eqnarray*} where $\mu$ is a finite signed Radon measure in $\mathrm{\Omega }$, $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded domain such that its complement ${\mathbb{R}}^n\mathrm{\backslash }\mathrm{\Omega }$ is uniformly $p$-thick and $\mathcal{A}$ is a Carath'eodory vector valued function satisfying growth and monotonicity conditions for the strongly singular case $1<p\leq \frac{3n-2}{2n-1}$. Our result extends the earlier results \cite{55Ph0,Tran19} to the strongly singular case $1<p\le \frac{3n-2}{2n-1}$ and a recent result \cite{HP} by considering rough conditions on the domain $\mathrm{\Omega }$ and the nonlinearity $\mathcal{A}$.