The compactness and the concentration compactness via p-capacity

Abstract

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 ${\mathcal {D}}^{1,p}_0(\Omega )$.