Beyond the Trudinger-Moser supremum

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Abstract

Let Ω be a bounded, smooth domain in ℝ 2. We consider the functional, in the supercritical Trudinger-Moser regime, i. e. for ∫ Ω{pipe}▼u{pipe} 2dx > 4π. More precisely, we are looking for critical points of I(u) in the class of functions u ε H 0 1(Ω) such that ∫ Ω{pipe}▼u{pipe} 2dx = 4π(1+α), for small α > 0. In particular, we prove the existence of 1-peak critical points of I(u) with ∫ Ω{pipe}▼u{pipe} 2dx = 4π(1+α) for any bounded domain Ω, 2-peak critical points with ∫ Ω{pipe}▼u{pipe} 2dx = 8π(1+α) for non-simply connected domains Ω, and k-peak critical points with ∫ Ω{pipe}▼u{pipe} 2dx = 4kπ(1+α) if Ω is an annulus.

Original languageEnglish
Pages (from-to)543-576
Number of pages34
JournalCalculus of Variations and Partial Differential Equations
Volume44
Issue number3-4
DOIs
Publication statusPublished - 1 Jul 2012

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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