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This paper concerns the formation of a coincidence set for the positivesolution of $p$-Laplacian elliptic problems of monostable type. It is provedthat for any small parameter of diffusion term, the solution coincides with thestable zero-function $a(x)$ of reaction term in an open set if $a(x)$ is$p$-harmonic (but, not constant) and a zero of order less than 1. Inversely, itis also shown that the solution is less than $a(x)$ if $a(x)$ is a zero oforder greater than or equal to 1. The proof rely on comparison theorems and anenergy method for obtaining local comaprison functions.

Coincidence sets in quasilinear elliptic problems of monostable type