Coincidence sets in quasilinear elliptic problems of monostable type

This paper concerns the formation of a coincidence set for the positive solution of the boundary value problem: -εΔ_pu=u^q-1f(a(x)-u) in ω with u=0 on ∂ ω, where ε is a positive parameter, Δ_pu=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 O_ε is empty for any ε>0. The proofs rely on comparison theorems and the energy method for obtaining local comparison functions.