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arXiv · 2506.19237

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

Kenichiro Umezu
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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 $\mu \to \infty$.

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Kenichiro Umezu. 2025-06-24. Boundary layer profiles of positive solutions for logistic equations with sublinear nonlinearity on the boundary. https://doi.org/10.1007/s00013-025-02161-7

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