Bubbling solutions for Moser-Trudinger type equations on compact Riemann surfaces

Pablo Figueroa, Monica Musso

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Abstract

We study an elliptic equation related to the Moser–Trudinger inequality on a compact Riemann surface (S,g), Δ gu+λ(ue u 2 −[Formula presented]∫Sue u 2 dv g)=0,in S,∫Sudv g=0, where λ>0 is a small parameter, |S| is the area of S, Δ g is the Laplace–Beltrami operator and dv g is the area element. Given any integer k≥1, under general conditions on S we find a bubbling solution u λ which blows up at exactly k points in S, as λ→0. When S is a flat two-torus in rectangular form, we find that either seven or nine families of such solutions do exist for k=2. In particular, in any square flat two-torus actually nine families of bubbling solutions with two bubbling points do exist. If S is a Riemann surface with non-constant Robin's function then at least two bubbling solutions with k=1 exists.

Original languageEnglish
Pages (from-to)2684-2739
Number of pages56
JournalJournal of Functional Analysis
Volume275
Issue number10
Early online date27 Aug 2018
DOIs
Publication statusPublished - 15 Nov 2018

Keywords

  • Green's function
  • Moser–Trudinger inequality

ASJC Scopus subject areas

  • Analysis

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