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

In this paper, we study the good-$\lambda$ type bounds for renormalized solutions 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$isafiniteRadonmeasureand$A$ is a monotone Carath'edory vector valued function defined on $W_0^{1,p}(\mathrm{\Omega })$. The operator $A$ satisfies growth and monotonicity conditions, and the $p$-capacity uniform thickness condition is imposed on ${\mathbb{R}}^n\setminus \mathrm{\Omega }$, for the singular case $\frac{3n-2}{2n-1}<p\le 2-\frac{1}{n}$. In fact, the same good-$\lambda$ type estimates were also studied by Quoc-Hung Nguyen and Nguyen Cong Phuc. For instance, in \cite{55QH4,55QH5}, authors’ method was also confined to the case of $\frac{3n-2}{2n-1}<p\le 2-\frac{1}{n}$ but under the assumption of $\mathrm{\Omega }$ is the Reifenberg flat domain and the coefficients of $A$ have small BMO (bounded mean oscillation) semi-norms. Otherwise, the same 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 account the case $\frac{3n-2}{2n-1}<p\le 2-\frac{1}{n}$ and without the hypothesis of Reifenberg flat domain on $\mathrm{\Omega }$ and small BMO semi-norms of $A$. Moreover, in rest of this paper, we also give the proof of the boundedness property of maximal function on Lorentz spaces and also the global gradient estimates of solution.