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On the existence of extremal functions in Sobolev embedding theorems with critical exponents

Author(s): A. V. Demyanov; A. I. Nazarov
Translated by: A. I. Nazarov
Original publication: Algebra i Analiz, tom 17 (2005), vypusk 5.
Journal: St. Petersburg Math. J. 17 (2006), 773-796.
MSC (2000): Primary 49J10, 35J20, 35J60
Posted: July 20, 2006
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Abstract | References | Similar articles | Additional information

Abstract: Sufficient conditions for the existence of extremal functions in Sobolev-type inequalities on manifolds with or without boundary are established. Some of these conditions are shown to be sharp. Similar results for embeddings in some weighted $ L_q$-spaces are obtained.

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Additional Information:

A. V. Demyanov
Affiliation: St. Petersburg State University, Russia
Email: alex@ad9503.spb.edu

A. I. Nazarov
Affiliation: St. Petersburg State University, Russia
Email: an@AN4751.spb.edu

DOI: 10.1090/S1061-0022-06-00929-0
PII: S 1061-0022(06)00929-0
Keywords: Minimizers, critical exponent, Sobolev inequality, Sobolev--Poincar\'e inequality, Hardy--Sobolev inequality, $p$-Laplacian
Received by editor(s): 30/NOV/2004
Posted: July 20, 2006
Additional Notes: Partially supported by the RF Ministry of Education (project no. 4733), and by RFBR (grant no. 05--01--01063).
Dedicated: In memory of Ol$\cprime$ga Aleksandrovna Ladyzhenskaya
Copyright of article: Copyright 2006, American Mathematical Society

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