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 ${\mathcal{D}}_0^{1,p}(\Omega )$ is the completion of $C_c^{\infty }(\Omega )$ with respect to the norm ${\left({\int }_{\Omega }|\nabla u{|}^p\right)}^{\frac{1}{p}}.$ 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 $L_{loc}^1(\Omega )$ that admits the following Hardy-Sobolev type inequality: \begin{eqnarray*} \int_Ω |g| |u|^p \leq C \int_Ω |\nabla u|^p, \forall\; u \in \mathcal{D}^{1,p}_0(Ω), \end{eqnarray*} for some $C>0.$ Further, we characterize the set of all $g$ in $\mathcal{H}(\Omega )$ for which the map $G(u)={\int }_{\Omega }g|u{|}^p$ is compact on ${\mathcal{D}}_0^{1,p}(\Omega )$. 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 ${\mathcal{D}}_0^{1,p}(\Omega )$.

27 pages, Changes in the hypothesis of Theorem 1.4 and Theorem 1.5