Global Bifurcation Theory for Fredholm Operator via Odd Crossing Number and Its Application

Abstract

We establish the Rabinowitz-type global bifurcation theorem and the Dancer-type unilateral global bifurcation theorem for $F(\lambda ,u)=0,(\lambda ,u)\in \mathbb{R}\times X$ , where $F:\mathbb{R}\times X\to Y$ is a $C^1$ map and locally proper with $F(\lambda ,0)=0$ for $\lambda \in \mathbb{R}$ , $X$ and $Y$ are real Banach spaces with $X\subseteq Y$ . Let $\mathcal{S}$ be the closure of the set of nontrivial solutions of $F(\lambda ,u)=0,(\lambda ,u)\in \mathbb{R}\times X$ . We shall show that, if $D_{u}F(\lambda ,u)$ is a Fredholm operator with index $0$ for all $(\lambda ,u)\in \mathbb{R}\times X$ and $D_{u}F(\lambda ,0)$ has an odd crossing number at $\lambda =\mu$ , then $\mathcal{S}$ possesses a maximal component ${\mathcal{C}}_{\mu }$ emanating from $(\mu ,0)$ , such that either ${\mathcal{C}}_{\mu }$ is unbounded or contains some $({\lambda }_{*},0)$ with ${\lambda }_{*}\ne \mu$ . Furthermore, if $\text{dim}\text{Ker}(D_{u}F(\mu ,0))=1$ for some $\mu \in \mathbb{R}$ , then $\mathcal{S}$ possesses two maximal sub-continua ${\mathcal{C}}_{\mu }^{\pm }$ emanating from $(\mu ,0)$ , such that either ${\mathcal{C}}_{\mu }^{+}$ and ${\mathcal{C}}_{\mu }^{-}$ are both unbounded or ${\mathcal{C}}_{\mu }^{+}\cap {\mathcal{C}}_{\mu }^{-}\ne \{(\mu ,0)\}$ . As one of applications, we obtain the unilateral global bifurcation result for an overdetermined elliptic problem. In addition, we give an example which satisfies all the assumptions of in our theorems but it doesn’t meet the classic Crandall–Rabinowitz transversality condition.