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