A Nonlinear elliptic PDE with curve singularity on the boundary
Mamadou Ciss, Abdourahmane Diatta, El Hadji Abdoulaye Thiam
TL;DR
The paper studies positive $H^1(\Omega)$-solutions to a Hardy–Sobolev trace problem with a curve singularity on the boundary. It develops a geometric–variational framework based on Fermi coordinates and metric expansions to capture boundary curvature effects, and analyzes a limiting half-space problem to obtain the critical constant $S_{N,s}$. Using carefully constructed test functions and concentration-compactness arguments, it establishes existence of minimizers for the Hardy–Sobolev trace constant $S_{\Omega,\Sigma,s}$ under two sharp criteria: a local curvature inequality and a global volume-based bound. The results hinge on precise asymptotics of the energy functional and its denominator, connecting boundary geometry to the attainability of the best constant. This advances understanding of elliptic problems with boundary singularities along curves and provides explicit geometric criteria for existence of solutions.
Abstract
Let $Ω$ be a bounded domain of $\mathbb{R}^{N+1}$ ($N \geq 3$) with smooth boundary $\partial Ω$ and $Σ$ be a closed submanifold contained on $\partial Ω$ and containing $0$. We are interesting in the existence of positive $H^1(Ω)$-solution of the following Hardy-Sobolev trace type equation \begin{equation*} \begin{cases} -Δu+u=0 \qquad & \textrm{ in $Ω$}\\\\ \displaystyle\frac{\partial u}{\partial ν}= ρ_Σ^{-s} u^{q_s-1} \qquad & \textrm{ on $\partial Ω$}, \end{cases} \end{equation*} where $ν$ is the unit outer normal of $\partial Ω$, $ρ_Σ: \partial Ω\to \mathbb{R}$ is the distance function in $\partial Ω$ to the curve $Σ$: $$ ρ_Σ(x):= \inf_{y \in Σ} d_{\tilde{g}}(x, y) $$ and for $0\leq s <1$, $q_s:=\frac{2(N-s)}{N-1}$ is the critical Hardy-Sobolev exponent. The existence of solution may depend on the local geometry of the boundary $\partial Ω$ and $Σ$ at $0$ or in the shapes of the domain $Ω$ and its boundary $\partial Ω$.