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On compactness in the Trudinger-Moser inequality
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2014 (English)In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. 13, no 2, 399-416 p.Article in journal (Refereed) Published
Abstract [en]

We show that the Moser functional J(u) = integral Omega(e(4 pi u2) - 1) dx on the set B = {u is an element of H-0(1)(Omega) : parallel to del u parallel to(2) <= 1}, where Omega subset of R-2 is a bounded domain, fails to be weakly continuous only in the following exceptional case. Define g(s)w(r) = s(-1/2)w(r(s)) for s > 0. If u(k) -> u in B while lim inf J(u(k)) > J(u), then, with some s(k) -> 0, u(k) = g(sk) [(2 pi)(-1/2) min {1, log1/vertical bar x vertical bar}], up to translations and up to a remainder vanishing in the Sobolev norm. In other words, the weak continuity fails only on translations of concentrating Moser functions. The proof is based on a profile decomposition similar to that of Solimini [16], but with different concentration operators, pertinent to the two-dimensional case.

Place, publisher, year, edition, pages
2014. Vol. 13, no 2, 399-416 p.
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URN: urn:nbn:se:uu:diva-231457ISI: 000339985500005OAI: oai:DiVA.org:uu-231457DiVA: diva2:744817
Available from: 2014-09-08 Created: 2014-09-08 Last updated: 2016-04-13

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