Coexisting steady-state solutions of a class of reaction-diffusion systems with different boundary conditions

In this work, we investigate a reaction-diffusion system in which both species are influenced by self-diffusion. Due to Hopf's boundary lemma, we obtain the boundedness of the classical solution of the system. By considering a particular function, we provide a complete characterization of the parameter ranges such that coexisting solutions of the system do not exist under three boundary conditions. Then based on the maximum principle, a sufficient condition for the existence of constant coexisting solutions of the system under Neumann boundary conditions was derived.