Bifurcation from infinity with applications to reaction-diffusion systems

The bifurcation method is one of powerful tools to study the existence of a continuous branch of solutions. However without further analysis, the local theory only ensures the existence of solutions within a small neighborhood of bifurcation point. In this paper we extend the theory of bifurcation from infinity, initiated by Rabinowitz [11] and Stuart [13], to find solutions of elliptic partial differential equations with large amplitude. For the applications to the reaction-diffusion systems, we are able to relax the conditions to obtain the bifurcation from infinity for the following nonlinear terms; (i) nonlinear terms satisfying conditions similar to [11] (all directions), (ii) nonlinear terms satisfying similar conditions only on the strip domain along the direction determined by the eigenfunction, (iii) p-homogeneous nonlinear terms with degenerate conditions.