Coincidence sets in quasilinear elliptic problems of monostable type

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This paper concerns the formation of a coincidence set for the positive solution of the boundary value problem: -εΔpu=uq-1f(a(x)-u) in ω with u=0 on ∂ ω, where ε is a positive parameter, Δpu=div(|∇u|p-2∇u), 1<q≤p<∞, f(s)~|s|θ-1s (s→0) for some θ>0 and a(x) is a positive smooth function satisfying Δpa=0 in ω with infω|∇a|>0. It is proved in this paper that if 0<θ<1 the coincidence set Oε={x∈ω:uε(x)=a(x)} has a positive measure for small ε and converges to ω with order O(ε1/p) as ε→0. Moreover, it is also shown that if θ≥1, then Oε is empty for any ε>0. The proofs rely on comparison theorems and the energy method for obtaining local comparison functions.

Original languageEnglish
Pages (from-to)2196-2208
Number of pages13
JournalJournal of Differential Equations
Issue number8
Publication statusPublished - 2011 Oct 15


  • Coincidence set
  • Dead core
  • Flat core
  • P-Laplacian

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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