## Abstract

This paper concerns the formation of a coincidence set for the positive solution of the boundary value problem: -εΔ_{p}u=u^{q-1}f(a(x)-u) in ω with u=0 on ∂ ω, where ε is a positive parameter, Δ_{p}u=div(|∇u|^{p-2}∇u), 1<q≤p<∞, f(s)~|s|^{θ-1}s (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 O_{ε} is empty for any ε>0. The proofs rely on comparison theorems and the energy method for obtaining local comparison functions.

Original language | English |
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Pages (from-to) | 2196-2208 |

Number of pages | 13 |

Journal | Journal of Differential Equations |

Volume | 251 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2011 Oct 15 |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics