On singular problems associated with mixed operators under mixed boundary conditions
Year: 2025
Type: preprint
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 Ω, u &= 0 \quad \text{in} \quad U^c, \mathcal{N}_s(u) &= 0 \quad \text{in} \quad \mathcal{N}, \frac{\partial u}{\partial ν} &= 0 \quad \text{in} \quad \partial Ω\cap \overline{\mathcal{N}}, \end{aligned} \right. \tag{$P_λ$} \end{equation*} where $U= (Ω\cup {\mathcal{N}} \cup (\partialΩ\cap\overline{\mathcal{N}}))$, $Ω\subseteq \mathbb{R}^N$ is a non empty open set, $\mathcal{D}$, $\mathcal{N}$ are open subsets of $\mathbb{R}^N\setminus{\bar{Ω}}$ such that ${\mathcal{D}} \cup {\mathcal{N}}= \mathbb{R}^N\setminus{\barΩ}$, $\mathcal{D} \cap {\mathcal{N}}= \emptyset $ and $Ω\cup \mathcal{N}$ is a bounded set with smooth boundary, $λ>0$ is a real parameter and $\mathcal{L}= -Δ+(-Δ)^{s},~ \text{for}~s \in (0, 1).$ Here $g(u)=u^{-q}$ or $g(u)= λu^{-q}+ u^p$ with $0
Source: arXiv (Cornell University)
Authors: Mukherjee, Tuhina, Sharma, Lovelesh
Institution: Indian Institute of Technology Jodhpur
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6948206964
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