On singular problems associated with mixed operators under mixed boundary conditions

TL;DR

This paper investigates singular boundary value problems involving mixed classical and fractional Laplace operators under mixed boundary conditions, establishing existence, regularity, and variational inequalities for solutions with singular nonlinearities.

Contribution

It introduces a novel analysis of singular problems with combined classical and fractional operators under mixed boundary conditions, providing existence, regularity, and variational insights.

Findings

Existence of weak solutions for the singular problems.

Solutions possess boundedness in the L-infinity norm.

Established Sobolev-type variational inequalities for solutions.

Abstract

In this paper, we study the following singular problem associated with mixed operators (the combination of the classical Laplace operator and the fractional Laplace operator) under mixed boundary conditions \begin{equation*} \label{1} \left\{ \begin{aligned} \mathcal{L}u &= g(u), \quad u > 0 \quad \text{in} \quad \Omega, u &= 0 \quad \text{in} \quad U^c, \mathcal{N}_s(u) &= 0 \quad \text{in} \quad \mathcal{N}, \frac{\partial u}{\partial \nu} &= 0 \quad \text{in} \quad \partial \Omega \cap \overline{\mathcal{N}}, \end{aligned} \right. \tag{$P_\lambda$} \end{equation*} where $U= (\Omega \cup {\mathcal{N}} \cup (\partial\Omega\cap\overline{\mathcal{N}}))$, $\Omega \subseteq \mathbb{R}^N$ is a non empty open set, $\mathcal{D}$, $\mathcal{N}$ are open subsets of $\mathbb{R}^N\setminus{\bar{\Omega }}$ such that ${\mathcal{D}} \cup {\mathcal{N}}= \mathbb{R}^N\setminus{\bar{\Omega}}$, $\mathcal{D} \cap {\mathcal{N}}= \emptyset $ and $\Omega\cup \mathcal{N}$ is a bounded set with smooth boundary, $\lambda >0$ is a real parameter and $\mathcal{L}= -\Delta+(-\Delta)^{s},~ \text{for}~s \in (0, 1).$ Here $g(u)=u^{-q}$ or $g(u)= \lambda u^{-q}+ u^p$ with $0<q<1<p\leq 2^*-1$. We study $(P_\lambda)$ to derive the existence of weak solutions along with its $L^\infty$-regularity. Moreover, some Sobolev-type variational inequalities associated with these weak solutions are established.