  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.
