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

We investigate the following mixed local and nonlocal quasilinear equation with singularity given by \begin{eqnarray*} \begin{split} -Δ_pu+(-Δ)q^s u&=\frac{f(x)}{u^δ}\text { in } Ω, \u&>0 $\text{ in }$ Ω,\u&=0 \text { in }$\mathbb{R}$^n \backslash Ω; \end{split} \end{eqnarray*} where, \begin{equation*} (-Δ)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 $\mathrm{\Omega }$ being a bounded domain in ${\mathbb{R}}^n$ with $C^2$ boundary, $1<q\le p<\mathrm{\infty }$, $s\in (0,1)$, $\delta >0$ and $f\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}(\mathrm{\Omega })$ is a non-negative function which behaves like $\mathbf{d}\mathbf{i}\mathbf{s}\mathbf{t}(\mathbf{x},\mathbf{\partial }\mathbf{\Omega }{)}^{-\mathbf{\beta }}$, $\beta \ge 0$ near $\mathrm{\partial }\mathrm{\Omega }$. We start by proving several Hölder and gradient Hölder regularity results for a more general class of quasilinear operators when $\delta =0$. Using the regularity results we deduce existence, uniqueness and Hölder regularity of a weak solution of the singular problem in $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}(\mathrm{\Omega })$ and its behavior near $\mathrm{\partial }\mathrm{\Omega }$ albeit with different exponents depending on $\beta +\delta$. Boundedness and Hölder regularity result to the singular equation with critical exponent were also discussed.