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

TL;DR

This paper establishes good-$mbda$ bounds for solutions to certain nonlinear elliptic equations with measure data, extending previous results to more general domains and coefficient conditions, and also proves related maximal function and gradient estimates.

Contribution

It extends good-$mbda$ bounds for quasilinear elliptic equations to cases without Reifenberg flatness and small BMO conditions, covering a broader class of singular problems.

Findings

Derived good-$mbda$ bounds for singular elliptic equations.

Proved boundedness of maximal functions on Lorentz spaces.

Established global gradient estimates for solutions.

Abstract

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.