Regularity results for a class of mixed local and nonlocal singular problems involving distance function

Abstract

We investigate the following mixed local and nonlocal quasilinear equation with singularity given by \begin{eqnarray*} \begin{split} -\Delta_pu+(-\Delta)_q^s u&=\frac{f(x)}{u^{\delta}}\text { in } \Omega, \\u&>0 \text{ in } \Omega,\\u&=0 \text { in }\mathbb{R}^n \backslash \Omega; \end{split} \end{eqnarray*} where, \begin{equation*} (-\Delta )_q^s u(x):= c_{n,s}\operatorname{P.V.}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+sq}} d y, \end{equation*} with $\Omega$ being a bounded domain in $\mathbb{R}^{n}$ with $C^2$ boundary, $1<q\leq p<\infty$, $s\in(0,1)$, $\delta>0$ and $f\in L^\infty_{\mathrm{loc}}(\Omega)$ is a non-negative function which behaves like $\mathbf{dist(x,\partial \Omega)^{-\beta}}$, $\beta\geq 0$ near $\partial \Omega$. We start by proving several H\"older and gradient H\"older regularity results for a more general class of quasilinear operators when $\delta=0$. Using the regularity results we deduce existence, uniqueness and H\"older regularity of a weak solution of the singular problem in $W_{\mathrm{loc}}^{1,p}(\Omega)$ and its behavior near $\partial \Omega$ albeit with different exponents depending on $\beta+\delta$. Boundedness and H\"older regularity result to the singular equation with critical exponent were also discussed.