  This paper concerns the formation of a coincidence set for the positive
solution of $p$-Laplacian elliptic problems of monostable type. It is proved
that for any small parameter of diffusion term, the solution coincides with the
stable 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, it
is also shown that the solution is less than $a(x)$ if $a(x)$ is a zero of
order greater than or equal to 1. The proof rely on comparison theorems and an
energy method for obtaining local comaprison functions.
