A sharp Hardy–Sobolev inequality with boundary term and applications

Hardy inequality for domains with a geometric boundary condition and applications

“…If $$\Omega \subset \mathbb {R}^N$$ is an arbitrary domain, Hardy–Sobolev inequalities and its variants have been the subject of intensive research, see [7, 15, 16, 23, 29] and references there in. For instance, Opic–Kufner [22] provide different conditions on the weight functions $$w_1$$ and $$w_2$$ for the validity of the Hardy–Sobolev inequality $$$$\begin{equation*} \int_{\mathrm{\Omega}}{w}_{1}{(x)|u|}^{p}\textit{dx}\leqslant \int_{\mathrm{\Omega}}{w}_{2}(x){|\nabla u|}^{p}\textit{dx},u\in {C}_{0}^{\infty}(\mathrm{\Omega}).$$…”

“…If Ω ⊂ ℝ 𝑁 is an arbitrary domain, Hardy-Sobolev inequalities and its variants have been the subject of intensive research, see [7,15,16,23,29] and references there in. For instance, Opic-Kufner [22] provide different conditions on the weight functions 𝑤 1 and 𝑤 2 for the validity of the Hardy-Sobolev inequality…”