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 $\Omega \subset \mathbb{R}^n$, $\mu$ is a finite Radon measure and $A$ is a monotone Carath\'edory vector valued function defined on $W^{1,p}_0(\Omega)$. The operator $A$ satisfies growth and monotonicity conditions, and the $p$-capacity uniform thickness condition is imposed on $\mathbb{R}^n \setminus \Omega$, 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 $\Omega$ 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 $\Omega$ 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.