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

TL;DR

This paper studies the boundary layer profiles of positive solutions to a logistic elliptic equation with sublinear boundary conditions, analyzing their asymptotic behavior as a parameter tends to infinity.

Contribution

It provides a detailed analysis of the asymptotic boundary layer profiles for positive solutions under sublinear Neumann boundary conditions in logistic equations.

Findings

Asymptotic profiles of solutions as parameter increases

Existence and uniqueness of positive solutions

Boundary layer behavior characterized for large parameter values

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 $\mu \to \infty$.