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

In this paper, we consider the logistic elliptic equation $-\mathrm{\Delta }u=u-u^p$ in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$, $N\ge 2$, equipped with the sublinear Neumann boundary condition $\frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }=\mu u^q$ on $\mathrm{\partial }\mathrm{\Omega }$, where $0<q<1<p$, and $\mu \ge 0$ 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 $\mu \to \mathrm{\infty }$.