Preprint

Inserted: 25 nov 2025
Last Updated: 25 nov 2025

Pages: 40
Year: 2025

Abstract:

We consider periodically perforated unbounded open sets and prove existence of extremals for the relevant sharp Poincaré-Sobolev embedding constant. The existence result holds no matter the shape or the regularity of the hole: it is sufficient that the latter is a compact set with positive capacity. We also show how to apply the main result in order to get a similar existence statement, for sets which are periodic in some directions and bounded in all the others.

Keywords: capacity, lack of compactness, Lane-Emden equation, Poincaré-Sobolev inequalities

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• brabripri_periodic_final.pdf