Stationary profiles of degenerate problems when a parameter is large

Abstract

The structure of positive solutions to nonlinear diffusion problems of the form $-\text{div}\ (|{\nabla} u|^{p-2}{\nabla} u) = \lambda f(u)$, in $\Omega$, $u = 0$ on $\partial \Omega$, $p > 1$, $\Omega \subset \mathbb R^N$ a bounded, smooth domain, is precisely studied as $\lambda \to +\infty $, for a class of logistic-type nonlinearities $f(u)$. By logistic it is understood that $f(u)/u^{p-1}$ is decreasing in $u > 0$, $f(u) \sim m u^{p-1}$, $m > 0$, as $u\to 0+$, while $f$ has a positive zero $u = u_0$ of order $k$. It is shown that the positive solution ${u_\lambda}$ homogenizes towards $u_0$ as $\lambda \to +\infty $, and develops a boundary layer near $\partial\Omega$ whose width is exactly measured. On the other hand, the arising of ``dead cores" $\{{u_\lambda} = u_0\}$ for $\lambda$ large is shown in the parameters regime $k < p-1$, the distance $\text{dist}(\{{u_\lambda} = u_0\},\partial\Omega)$ to $\partial \Omega$ being also exactly estimated as $\lambda \to +\infty $. Thus, earlier results in [12], [22] are substantially sharpened. In addition, suitable lower-order perturbations at infinity of the problem are studied.