The compactness and the concentration compactness via p -capacity
2021
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 $${\mathcal {D}}^{1,p}_0(\Omega )$$
.
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