Logistic elliptic equation with a nonlinear boundary condition arising from coastal fishery harvesting

TL;DR

This paper studies positive solutions of a logistic elliptic equation with a nonlinear boundary condition modeling coastal fishery harvesting, revealing existence and multiplicity results depending on parameters and ecological interpretations.

Contribution

It introduces a new boundary condition derived from fishery harvesting and analyzes the existence and multiplicity of solutions based on parameter regimes.

Findings

Multiple positive solutions exist for small harvesting intensity when a key eigenvalue exceeds one.

No positive solutions exist for large harvesting intensity in certain parameter regimes.

At least one positive solution exists for all harvesting intensities when the eigenvalue is below one.

Abstract

Let $0<q<1<p$. In this study, we investigate positive solutions of the logistic elliptic equation $-\Delta u = u(1-u^{p-1})$ in a smooth bounded domain $\Omega$ of $\mathbb{R}^N$, $N\geq1$, with the nonlinear boundary condition $\frac{\partial u}{\partial \nu}=-\lambda u^q$ on $\partial\Omega$. This nonlinear boundary condition arises from coastal fishery harvesting. When $p>1$ is subcritical, we prove that in the case of $\lambda_{\Omega}>1$, there exist at least two positive solutions for $\lambda>0$ sufficiently small but no positive solutions for $\lambda>0$ large enough. In the case of $\lambda_{\Omega}<1$, there exists at least one positive solution for every $\lambda>0$. Here, $\lambda_{\Omega}>0$ is the smallest eigenvalue of $-\Delta$ under the Dirichlet boundary condition. An interpretation of our main results from an ecological viewpoint is presented.