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

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.