Method of Monotone Solutions for Reaction-Diffusion Equations

Existence of solutions of reaction-diffusion systems of equations in unbounded domains is studied by the Leray–Schauder (LS) method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces. We identify some reaction-diffusion systems for which there exist two subclasses of solutions separated in the function space, monotone and nonmonotone solutions. A priori estimates and existence of solutions are obtained for monotone solutions allowing to prove their existence by the LS method. Various applications of this method are given. © 2021, Springer Science+Business Media, LLC, part of Springer Nature.