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

Let $\mathrm{\Omega }$ be a smooth bounded domain of ${\mathbb{R}}^{N+1}$ of boundary $\mathrm{\partial }\mathrm{\Omega }={\mathrm{\Gamma }}_1\cup {\mathrm{\Gamma }}_2$ and such that $\partial \Omega \cap \Gamma_2$isaneighborhoodof$0$,$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} \Delta u= 0& \qquad $\text{ in }$ \Omega\\ u=0 & \qquad $\text{ on }$ \Gamma_1 \\ \frac{\partial u}{\partial \nu}=h(x) u + \frac{u^{q(s)-1}}{d(x)^{s}} & \qquad $\text{ on }$ \Gamma_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 $\partial \Omega$\mathrm{.}Inparticular,weproveexistenceofminimizerswhen$N \geq 3$ and the mean curvature is sufficiently below the potential $h$ at $0$.