Coincidence sets in quasilinear elliptic problems of monostable type

TL;DR

This paper investigates the formation of coincidence sets in p-Laplacian elliptic problems of monostable type, revealing how the solution's behavior depends on the properties of the reaction term's zero set.

Contribution

It establishes conditions under which solutions coincide with or are less than the reaction term, based on the harmonicity and order of zeros of the reaction function.

Findings

Solutions coincide with the zero function in certain conditions.

Solutions are less than the reaction term when zeros have higher order.

The results depend on the p-harmonicity and zero order of the reaction term.

Abstract

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.