Boundary layer profiles of positive solutions for logistic equations with sublinear nonlinearity on the boundary

Abstract

In this paper, we consider the logistic elliptic equation $-\Delta u = u- u^{p}$ in a smooth bounded domain $\Omega \subset \mathbb{R}^{N}$, $N\geq2$, equipped with the sublinear Neumann boundary condition $\frac{\partial u}{\partial \nu} = \mu u^{q}$ on $\partial \Omega$, where $0<q<1<p$, and $\mu\geq0$ is a parameter. With sub- and super-solutions and a comparison principle for the equation, we analyze the asymptotic profile of a unique positive solution for the equation as $\mu \to \infty$.