The logistic elliptic equation with the nonlinear boundary condition arising from coast fishery harvesting

Abstract

Let 0 < q < 1 < p. In this study, we investigate positive solutions of the logistic elliptic equation −∆u = u(1 − u p−1 ) in a smooth bounded domain Ω of R N , N ≥ 1, with the nonlinear boundary condition ∂u ∂ν = −λu q on ∂Ω. This nonlinear boundary condition arises from coast fishery harvesting. When p > 1 is subcritical, we prove that in the case of λΩ > 1, 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, there exists at least one positive solution for every λ > 0. Here, λΩ > 0 is the smallest eigenvalue of −∆ under the Dirichlet boundary condition. An interpretation of our main results from an ecological viewpoint is presented.