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 −Δu=u(1−up−1) in a smooth bounded domain Ω of RN, N≥1, with the nonlinear boundary condition ∂u∂ν=−λuq on ∂Ω. This nonlinear boundary condition arises from coastal fishery harvesting. We prove that there exists at least one positive solution for every λ>0 in the case of λΩ<1. Furthermore, when p>1 is subcritical, we prove that there exist at least two positive solutions for λ>0 sufficiently small but no positive solutions for λ>0 large enough in the case of λΩ>1. Here, λΩ>0 is the smallest eigenvalue of −Δ under the Dirichlet boundary condition. An interpretation of our main results from an ecological viewpoint is presented.