On singular problems associated with mixed operators under mixed boundary conditions
Tuhina Mukherjee, Lovelesh Sharma
TL;DR
This work analyzes a singular elliptic problem driven by a mixed local-nonlocal operator $\mathcal{L}=-\Delta+(-\Delta)^s$ under mixed Dirichlet-Neumann boundary conditions, with nonlinearities $g(u)=u^{-q}$ or $g(u)=\lambda u^{-q}+u^p$, in a bounded smooth domain. It develops a robust variational framework using the space $\mathcal{X}^{1,2}_{\mathcal{D}}(U)$ and a mixed Sobolev inequality, proves existence and uniqueness for the purely singular case, and establishes $L^{\infty}$-regularity of solutions. For the perturbed problem, it employs a Nehari-manifold approach to obtain two positive weak solutions for small $\lambda$, along with thorough regularity and comparison results. The results extend the theory of mixed local-nonlocal operators under mixed boundary conditions to singular and near-critical nonlinearities, providing foundational tools and sharp inequalities with potential applications in nonlinear and nonlocal PDE contexts.
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 $0<q<1<p\leq 2^*-1$. We study $(P_λ)$ 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.