  The aim of this paper is to present the global estimate for gradient of
renormalized solutions to the following quasilinear elliptic problem:
\begin{align*} \begin{cases} -div(A(x,\nabla u)) &= \mu \quad \text{in} \ \
\Omega, \\ u &=0 \quad \text{on} \ \ \partial \Omega, \end{cases} \end{align*}
in Lorentz-Morrey spaces, where $\Omega \subset \mathbb{R}^n$ ($n \ge 2$),
$\mu$ is a finite Radon measure, $A$ is a monotone Carath\'eodory vector valued
function defined on $W^{1,p}_0(\Omega)$ and the $p$-capacity uniform thickness
condition is imposed on the complement of our domain $\Omega$. It is remarkable
that the local gradient estimates has been proved firstly by G. Mingione in
\cite{Mi3} at least for the case $2 \le p \le n$, where the idea for extending
such result to global ones was also proposed in the same paper. Later, the
global Lorentz-Morrey and Morrey regularities were obtained by N.C.Phuc in
\cite{55Ph1} for regular case $p>2 - \frac{1}{n}$. Here in this study, we
particularly restrict ourselves to the singular case $\frac{3n-2}{2n-1}<p\le
2-\frac{1}{n}$. The results are central to generalize our technique of
good-$\lambda$ type bounds in previous work \cite{MP2018}, where the local
gradient estimates of solution to this type of equation was obtained in the
Lorentz spaces. Moreover, the proofs of most results in this paper are
formulated globally up to the boundary results.
