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

TL;DR

This paper analyzes positive solutions of a logistic elliptic equation with nonlinear boundary conditions modeling coastal fishery harvesting, extending previous work to the critical case and refining the solution set description.

Contribution

It extends the analysis of positive solutions to the critical eigenvalue case and provides a more detailed characterization of the solution set.

Findings

Extended analysis to the critical eigenvalue case.

Refined description of positive solution set.

Applied energy method, sub- and supersolutions, and implicit function analysis.

Abstract

We study the positive solutions of the logistic elliptic equation with a nonlinear Neumann boundary condition that models coastal fishery harvesting ([18]). An essential role is played by the smallest eigenvalue of the Dirichlet eigenvalue problem, with respect to which a noncritical case is studied in [32]. In this paper, we extend our analysis to the critical case and further study the noncritical case for a more precise description of the positive solution set. Our approach relies on the energy method, sub- and supersolutions, and implicit function analysis.