The compactness and the concentration compactness via p-capacity

• Published in: Annali di matematica pura ed applicata 2021, Vol.200 (6), p.2715-2740
• Main Authors: Anoop, T. V., Das, Ujjal
• Format: Article
• Language: English
• Subjects: Function space; Mathematics; Mathematics and Statistics
• ISSN: 0373-3114; 1618-1891
• DOI: 10.1007/s10231-021-01098-2

For p ∈ ( 1 , N ) and Ω ⊆ R N open, the Beppo-Levi space D 0 1 , p ( Ω ) is the completion of C c ∞ ( Ω ) with respect to the norm ∫ Ω | ∇ u | p d x 1 p . Using the p -capacity, we define a norm and then identify the Banach function space H ( Ω ) with the set of all g in L loc 1 ( Ω ) that admits the following Hardy–Sobolev type inequality: ∫ Ω | g | | u | p d x ≤ C ∫ Ω | ∇ u | p d x , ∀ u ∈ D 0 1 , p ( Ω ) , for some C > 0 . Further, we characterize the set of all g in H ( Ω ) for which the map G ( u ) = ∫ Ω g | u | p d x is compact on D 0 1 , p ( Ω ) . We use a variation of the concentration compactness lemma to give a sufficient condition on g ∈ H ( Ω ) so that the best constant in the above inequality is attained in D 0 1 , p ( Ω ) .