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 $\left( \int_{\Omega}|\nabla u|^p \right)^ \frac{1}{p}.$ 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{eqnarray*} \int_{\Omega} |g| |u|^p \leq C \int_{\Omega} |\nabla u|^p, \forall\; u \in \mathcal{D}^{1,p}_0(\Omega), \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 $\mathcal{D}^{1,p}_0(\Omega)$. 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 $\mathcal{D}^{1,p}_0(\Omega)$.