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

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