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In this paper, we study the good-$\lambda$ type bounds for renormalizedsolutions to nonlinear 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*} where $\Omega \subset\mathbb{R}^n$, $\mu$ is a finite Radon measure and $A$ is a monotoneCarath\'edory vector valued function defined on $W^{1,p}_0(\Omega)$. Theoperator $A$ satisfies growth and monotonicity conditions, and the $p$-capacityuniform thickness condition is imposed on $\mathbb{R}^n \setminus \Omega$, forthe singular case $\frac{3n-2}{2n-1} < p \le 2- \frac{1}{n}$. In fact, the samegood-$\lambda$ type estimates were also studied by Quoc-Hung Nguyen and NguyenCong Phuc. For instance, in \cite{55QH4,55QH5}, authors' method was alsoconfined to the case of $\frac{3n-2}{2n-1} < p \le 2- \frac{1}{n}$ but underthe assumption of $\Omega$ is the Reifenberg flat domain and the coefficientsof $A$ have small BMO (bounded mean oscillation) semi-norms. Otherwise, thesame problem was considered in \cite{55Ph0} in the regular case of$p>2-\frac{1}{n}$. In this paper, we extend their results, taking into accountthe case $\frac{3n-2}{2n-1} < p \le 2- \frac{1}{n}$ and without the hypothesisof Reifenberg flat domain on $\Omega$ and small BMO semi-norms of $A$.Moreover, in rest of this paper, we also give the proof of the boundednessproperty of maximal function on Lorentz spaces and also the global gradientestimates of solution.

Good-$\lambda$ type bounds of quasilinear elliptic equations for the singular case