A Nonlinear elliptic PDE with curve singularity on the boundary

TL;DR

This paper investigates the existence of positive solutions to a nonlinear elliptic PDE with a boundary singularity along a curve, influenced by the local geometry of the domain and boundary.

Contribution

It establishes conditions under which positive solutions exist for a Hardy-Sobolev trace problem with boundary curve singularity, considering geometric influences.

Findings

Existence depends on local boundary and curve geometry.

Solutions exist under specific geometric and domain conditions.

The problem involves critical Hardy-Sobolev exponents.

Abstract

Let $\Omega$ be a bounded domain of $\mathbb{R}^{N+1}$ ($N \geq 3$) with smooth boundary $\partial \Omega$ and $\Sigma$ be a closed submanifold contained on $\partial \Omega$ and containing $0$. We are interesting in the existence of positive $H^1(\Omega)$-solution of the following Hardy-Sobolev trace type equation \begin{equation*} \begin{cases} -\Delta u+u=0 \qquad & \textrm{ in $\Omega$}\\\\ \displaystyle\frac{\partial u}{\partial \nu}= \rho_{\Sigma}^{-s} u^{q_s-1} \qquad & \textrm{ on $\partial \Omega$}, \end{cases} \end{equation*} where $\nu$ is the unit outer normal of $\partial \Omega$, $\rho_\Sigma: \partial \Omega \to \mathbb{R}$ is the distance function in $\partial \Omega$ to the curve $\Sigma$: $$ \rho_\Sigma(x):= \inf_{y \in \Sigma} d_{\tilde{g}}(x, y) $$ 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 $\partial \Omega$ and $\Sigma$ at $0$ or in the shapes of the domain $\Omega$ and its boundary $\partial \Omega$.