Bubbling along boundary geodesics near the second critical exponent

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30 Citations (Scopus)

Abstract

The role of the second critical exponent p = (n + 1)/(n - 3), the Sobolev critical exponent in one dimension less, is investigated for the classical Lane-Emden-Fowler problem Δu + up = 0, u > 0 under zero Dirichlet boundary conditions, in a domain Ω in ℝn with bounded, smooth boundary. Given Γ, a geodesic of the boundary with negative inner normal curvature we find that for p = (n + 1)/(n - 3) - ε, there exists a solution uε such that |∇u ε|2converges weakly to a Dirac measure on Γ as ε → 0+, provided that γ is nondegenerate in the sense of second variations of length and ε remains away from a certain explicit discrete set of values for which a resonance phenomenon takes place.

Original languageEnglish
Pages (from-to)1553-1605
Number of pages53
JournalJournal of the European Mathematical Society
Volume12
Issue number6
DOIs
Publication statusPublished - 28 Sep 2010

Keywords

  • Blowing-up solution
  • Critical Sobolev exponent
  • Nondegenerate geodesic

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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