| Contents |
| We prove that for $N \geq 5$, $\Omega \subset \mathbb{R}^N$ a bounded domain with boundary smooth, $ 1 \leq q < \frac{N}{N-4}$ and $0 \leq \tau < 4$, the critical Hardy-Sobolev constant $C_{HS}$ does not depend on all the traces of the space $H^2(\Omega)$, i.e. we prove that the critical Hardy-Sobolev constant with Navier conditions coincides with the constant with Dirichlet conditions.
Moreover, in the same assumptions, we prove two improvements of the Hardy-Sobolev inequality, with Navier and with Dirichlet boundary conditions. In the first case we prove that there exists a positive constant $C=C(N,q,\tau)$ such that the following critical Hardy-Sobolev inequality holds for any $u \in H^2_\vartheta(\Omega)$ and for any $1 \leq q < \frac{N}{N-4}$
$$
\int_\Omega |\Delta u|^2 dx \geq C_{HS} \left(\int_\Omega \frac{|u|^{\sigma}}{|x|^\tau} dx\right)^{\frac{2}{\sigma}} + C\left(\int_\Omega |u|^q dx\right)^{\frac{2}{q}},
$$
with $C_{HS}=C_{HS}(\tau)$ the critical Hardy-Sobolev constant.
In the second case we prove that there exists a positive constant $C=C(N,q,\tau)$ such that for any $u \in H^2_0(\Omega)$ and for any $1 \leq q \leq \frac{N}{N-4}$
$$
\int_\Omega |\Delta u|^2 dx \geq C_{HS} \left(\int_\Omega \frac{|u|^{\sigma}}{|x|^\tau} dx\right)^{\frac{2}{\sigma}} + C||u||_{L^q_w(\Omega)}^2.
$$
Both results are sharp in $q$. |
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