Flat core properties associated to the $p$-Laplace operator

Abstract:

We study the formation of a flat hat pattern in the profile of the positive solution of an equation of the type: $\varepsilon {\Delta _p}u - {u^{p - 1}}{(1 - u)^\theta } = 0\;(0 < \theta < p - 1)$ in a bounded domain $\Omega$. When $\varepsilon$ tends to ${0^ + }$, the growth of the zone where $u = {u_\varepsilon }$ takes the value $1$ in $\Omega$ is studied.