Global Lorentz gradient estimates for quasilinear equations with measure data for the strongly singular case: $1<p\leq \frac{3n-2}{2n-1}$

Abstract

In this paper, we study the global regularity estimates in Lorentz spaces for gradients of solutions to quasilinear elliptic equations with measure data of the form \begin{eqnarray*} \left\{ \begin{array}{rcl} -{\rm div}(\mathcal{A}(x, \nabla u))&=& \mu \quad \text{in} ~\Omega, u&=&0 \quad \text{on}~ \partial \Omega, \end{array}\right. \end{eqnarray*} where $\mu$ is a finite signed Radon measure in $\Omega$, $\Omega \subset \mathbb{R}^n$ is a bounded domain such that its complement $\mathbb{R}^n\backslash\Omega$ is uniformly $p$-thick and $\mathcal{A}$ is a Carath\'eodory vector valued function satisfying growth and monotonicity conditions for the strongly singular case $1<p\leq \frac{3n-2}{2n-1}$. Our result extends the earlier results \cite{55Ph0,Tran19} to the strongly singular case $1<p\leq \frac{3n-2}{2n-1}$ and a recent result \cite{HP} by considering rough conditions on the domain $\Omega$ and the nonlinearity $\mathcal{A}$.