The compactness and the concentration compactness via p-capacity

For $$p \in (1,N)$$ and $$\Omega \subseteq {\mathbb {R}}^N$$ open, the Beppo-Levi space $${\mathcal {D}}^{1,p}_0(\Omega )$$ is completion of $$C_c^{\infty }(\Omega with respect to norm $$\left[ \int _{\Omega }|\nabla u|^p dx \right] ^ \frac{1}{p}.$$ Using p-capacity, we define a then identify Banach function {H}}(\Omega set all g in $$L^1_{loc}(\Omega that admits following Hardy–Sobolev type inequality: $$\begin{aligned} } |g| |u|^p \le C |\nabla dx, \forall \; u {\mathcal ), \end{aligned}$$ for some $$C>0.$$ Further, characterize which map $$G(u)= \displaystyle dx$$ compact on . We use variation concentration compactness lemma give sufficient condition $$g\in so best constant above inequality attained