Diffusive logistic equation with a non-Lipschitz nonlinear boundary condition arising from coastal fishery harvesting: the resonant case

For bifurcation analysis, we study the positive solution set for a semilinear elliptic equation of the logistic type, equipped with a sublinear boundary condition modeling coastal fishery harvesting. This work is a continuation of the author’s previous studies (Umezu in Nonlinear Anal. Real World Appl. 70:103788, 2023, J. Math. Anal. Appl. 534:128134, 2024), where certain results were obtained in a non-resonant case, including the existence, uniqueness, multiplicity, and strong positivity for positive solutions. In this paper, we consider the delicate resonant case and develop a sort of nonstandard bifurcation technique at zero to evaluate the positive solution set depending on a parameter. The nonlinear boundary condition is not right-differentiable at zero.