Sobolev and H\"older regularity results for some singular nonhomogeneous quasilinear problems

Abstract

This article deals with the study of the following singular quasilinear equation: \begin{equation*} (P) \left\{ \ -\Delta_{p}u -\Delta_{q}u = f(x) u^{-\delta},\; u>0 \text{ in }\; \Om; \; u=0 \text{ on } \pa\Om, \right. \end{equation*} where $\Om$ is a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary $\pa\Om$, $1< q< p<\infty$, $\de>0$ and $f\in L^\infty_{loc}(\Om)$ is a non-negative function which behaves like $\textnormal{dist}(x,\pa\Om)^{-\ba},$ $\ba\ge 0$ near the boundary of $\Om$. We prove the existence of a weak solution in $W^{1,p}_{loc}(\Om)$ and its behaviour near the boundary for $\ba<p$. Consequently, we obtain optimal Sobolev regularity of weak solutions. By establishing the comparison principle, we prove the uniqueness of weak solution for the case $\ba<2-\frac{1}{p}$. Subsequently, for the case $\ba\ge p$, we prove the non-existence result. Moreover, we prove H\"older regularity of the gradient of weak solution to a more general class of quasilinear equations involving singular nonlinearity as well as lower order terms (see \eqref{Prb}). This result is completely new and of independent interest. In addition to this, we prove H\"older regularity of minimal weak solutions of $(P)$ for the case $\beta+\delta\geq 1$ that has not been fully answered in former contributions even for $p$-Laplace operators.