A Nonlinear elliptic PDE with curve singularity on the boundary

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 $\partial \Omega$.