A Nonlinear elliptic PDE with curve singularity on the boundary

Abstract

Let $\hat{I}\text{c◯}$ be a bounded domain of ${\mathbb{R}}^{N+1}$ ($N\ge 3$) with smooth boundary $\partial \hat{I}\text{c◯}$ and $\hat{I}\pounds$ be a closed submanifold contained on $\partial \hat{I}\text{c◯}$ and containing $0$. We are interesting in the existence of positive $H^1(\hat{I}\text{c◯})$-solution of the following Hardy-Sobolev trace type equation \begin{equation*} \begin{cases} -Δu+u=0 \qquad & \textrm{ in $\hat{I}\text{c◯}$}\\\\ \displaystyle\frac{\partial u}{\partial ν}= ρ_Σ^{-s} u^{q_s-1} \qquad & \textrm{ on $\partial \hat{I}\text{c◯}$}, \end{cases} \end{equation*} where $\hat{I}½$ is the unit outer normal of $\partial \hat{I}\text{c◯}$, $\ddot{I}{}_{\hat{I}}\pounds :\partial \hat{I}\text{c◯}\to \mathbb{R}$ is the distance function in $\partial \hat{I}\text{c◯}$ to the curve $\hat{I}\pounds$: $$\ddot{I}{}_{\hat{I}}\pounds (x):=\underset{y\in \hat{I}\pounds }{\inf }{d}_{\tilde{g}}(x,y)$$ and for $0\le s<1$, $q_s:=\frac{2(N-s)}{N-1}$ is the critical Hardy-Sobolev exponent. The existence of solution may depend on the local geometry of the boundary $\partial \hat{I}\text{c◯}$ and $\hat{I}\pounds$ at $0$ or in the shapes of the domain $\hat{I}\text{c◯}$ and its boundary $\partial \hat{I}\text{c◯}$.