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Czechoslovak Mathematical Journal

, Volume 65, Issue 2, pp 493–516 | Cite as

Concentration-compactness principle for embedding into multiple exponential spaces on unbounded domains

  • Robert Černý
Article

Abstract

Let Ω ⊂ ℝ n be a domain and let α < n − 1. We prove the Concentration-Compactness Principle for the embedding of the space W 0 1 L n log α L(Ω) into an Orlicz space corresponding to a Young function which behaves like (t n/(n−1−α)) for large t. We also give the result for the embedding into multiple exponential spaces.

Our main result is Theorem 1.6 where we show that if one passes to unbounded domains, then, after the usual modification of the integrand in the Moser functional, the statement of the Concentration-Compactnes Principle is very similar to the statement in the case of a bounded domain. In particular, in the case of a nontrivial weak limit the borderline exponent is still given by the formula
$$P: = \left( {1 - \left\| {\Phi (|\nabla u|)} \right\|_{L^1 (\mathbb{R}^n )} } \right)^{ - 1/(n - 1)} $$
.

Keywords

Sobolev space Orlicz-Sobolev space Moser-Trudinger inequality sharp constant concentration-compactness principle 

MSC 2010

46E35 46E30 26D10 

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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2015

Authors and Affiliations

  1. 1.Department of Mathematical Analysis, Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic

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