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

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