Coincidence sets in quasilinear elliptic problems of monostable type

研究成果: Article査読

抄録

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.

本文言語English
ページ(範囲)2196-2208
ページ数13
ジャーナルJournal of Differential Equations
251
8
DOI
出版ステータスPublished - 2011 10 15

ASJC Scopus subject areas

  • 分析
  • 応用数学

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