Bifurcation Theory for Fredholm Operators

This paper consists of four parts. It begins by using the authors's generalized Schauder formula, \cite{JJ}, and the algebraic multiplicity, $\chi$, of Esquinas and Lopez-Gomez \cite{ELG,Es,LG01} to package and sharpening all existing results in local and global bifurcation theory for Fredholm operators through the recent author's axiomatization of the Fitzpatrick--Pejsachowicz--Rabier degree, \cite{JJ2}. This facilitates reformulating and refining all existing results in a compact and unifying way. Then, the local structure of the solution set of $\mathfrak{F}(\lambda,u)=0$ at a simple degenerate eigenvalue is ascertained by means of some concepts and devices of Algebraic Geometry and Galois Theory, which establishes a bisociation between Bifurcation Theory and Algebraic Geometry. Further, we combine the theorem of structure of analytic manifolds with a brilliant idea of Buffoni and Toland \cite{BT} to show that the solution sets of the most paradigmatic one-dimensional boundary value problems with analytic nonlinearities actually consist of global analytic arcs of curve. Finally, the unilateral theorems of \cite{LG01,LG02}, as well as the refinement of Xi and Wang \cite{XW}, are substantially generalized. This paper also analyzes two important examples to illustrate and discuss the relevance of the abstract theory. The second one studies the regular positive solutions of a multidimensional quasilinear boundary value problem of mixed type related to the mean curvature operator.