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} -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 $Ω$ being a bounded domain in $\mathbb{R}^{n}$ with $C^2$ boundary, $1 0$ and $f\in L^\infty_{\mathrm{loc}}(Ω)$ is a non-negative function which behaves like $\mathbf{dist(x,\partial Ω)^{-β}}$, $β\geq 0$ near $\partial Ω$. We start by proving several Hölder and gradient Hölder regularity results for a more general class of quasilinear operators when $δ=0$. Using the regularity results we deduce existence, uniqueness and Hölder regularity of a weak solution of the singular problem in $W_{\mathrm{loc}}^{1,p}(Ω)$ and its behavior near $\partial Ω$ albeit with different exponents depending on $β+δ$. Boundedness and Hölder regularity result to the singular equation with critical exponent were also discussed.