Regularity results for a class of mixed local and nonlocal singular problems involving distance function
Kaushik Bal, Stuti Das
TL;DR
This work analyzes a mixed local-nonlocal quasilinear equation $-\Delta_p u + (-\Delta)^s_q u = f(x) u^{-\delta}$ in a bounded $C^2$ domain $\Omega$, with $u>0$ in $\Omega$ and $u=0$ outside, where $f(x) \sim \operatorname{dist}(x,\partial\Omega)^{-\beta}$. It develops a comprehensive regularity theory for both regular and singular problems, including local Hölder and gradient Hölder estimates, up-to-boundary regularity, and optimal Sobolev regularity, using perturbation, barrier, and tail techniques. It proves existence, uniqueness, and precise boundary behavior for the singular problem, obtains optimal Sobolev exponents and Hölder up to the boundary with several regime-dependent cases, and shows a nonexistence result when $\beta\ge p$. An application to a perturbed problem yields $C^{1,\alpha}(\overline{\Omega})$ regularity, highlighting how operator nonhomogeneity and the double singularity influence regularity. Overall, the results provide a sharp, regime-sensitive regularity theory for a broad class of nonhomogeneous mixed local-nonlocal singular problems with boundary blow-up weights.
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, $1<q\leq p<\infty$, $s\in(0,1)$, $δ>0$ 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.