The role of the mean curvature in a mixed Hardy-Sobolev trace inequality

Abstract

Let $\Omega$ be a smooth bounded domain of ${\mathbb{R}}^{N+1}$ of boundary $\partial \Omega ={\Gamma }_1\cup {\Gamma }_2$ and such that $\partial \Omega \cap {\Gamma }_2$ is a neighborhood of $0$, $h\in {\mathcal{C}}^0(\partial \Omega \cap {\Gamma }_2)$ and $s\in [0,1)$. We propose to study existence of positive solutions to the following Hardy-Sobolev trace problem with mixed boundaries conditions \begin{align} \begin{cases} Δu= 0& \qquad \textrm{ in } Ω\\\ u=0 & \qquad \textrm{ on } Γ_1 \\\ \frac{\partial u}{\partial ν}=h(x) u + \frac{u^{q(s)-1}}{d(x)^{s}} & \qquad \textrm{ on } Γ_2, \end{cases} \end{align} where $q(s):=\frac{2(N-s)}{N-1}$ is the critical Hardy-Sobolev trace exponent and $\nu$ is the outer unit normal of $\partial \Omega$. In particular, we prove existence of minimizers when $N\ge 3$ and the mean curvature is sufficiently below the potential $h$ at $0$.

arXiv admin note: text overlap with arXiv:1411.4414