Universal potential estimates for $ 1 < p\leq 2-\frac{1}{n} $

Abstract

We extend the so-called universal potential estimates of Kuusi-Mingione type (J. Funct. Anal. 262: 4205–4269, 2012) to the singular case $ 1 < p\leq 2-1/n $ for the quasilinear equation with measure data

\begin{document}$ \begin{equation*} -\operatorname{div}(A(x,\nabla u)) = \mu \end{equation*} $\end{document}

in a bounded open subset $ \Omega $ of $ \mathbb{R}^n $, $ n\geq 2 $, with a finite signed measure $ \mu $ in $ \Omega $. The operator $ \operatorname{div}(A(x, \nabla u)) $ is modeled after the $ p $-Laplacian $ \Delta_p u: = {\rm div}\, (|\nabla u|^{p-2}\nabla u) $, where the nonlinearity $ A(x, \xi) $ ($ x, \xi \in \mathbb{R}^n $) is assumed to satisfy natural growth and monotonicity conditions of order $ p $, as well as certain additional regularity conditions in the $ x $-variable.