Global gradient estimates for very singular quasilinear elliptic equations with measure data

This paper continues the development of regularity results for quasilinear measure data problems \begin{align*} \begin{cases} -\mathrm{div}(A(x,\nabla u)) &= \mu \quad \text{in} \ \ \Omega, \\ \quad \quad \qquad u &=0 \quad \text{on} \ \ \partial \Omega, \end{cases} \end{align*} in Lorentz and Lorentz-Morrey spaces, where $\Omega \subset \mathbb{R}^n$ ($n \ge 2$), $\mu$ is a finite Radon measure on $\Omega$, and $A$ is a monotone Carath\'eodory vector valued operator acting between $W^{1,p}_0(\Omega)$ and its dual $W^{-1,p'}(\Omega)$. It emphasizes that this paper studies the `very singular' case $1<p \le \frac{3n-2}{2n-1}$ and the problem is considered under the weak assumption, where the $p$-capacity uniform thickness condition is imposed on the complement of domain $\Omega$. There are two main results obtained in our study pertaining to the global gradient estimates of solutions in Lorentz and Lorentz-Morrey spaces involving the use of maximal and fractional maximal operators. The idea for writing this working paper comes directly from the recent results by others in the same research topic, where global estimates for gradient of solutions for the `very singular' case still remains a challenge, specifically related to Lorentz and Lorentz-Morrey spaces.