Bifurcation from infinity with applications to reaction-diffusion systems

Abstract

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) \begin{document}$ p $\end{document} -homogeneous nonlinear terms with degenerate conditions.