Bifurcation theory for Fredholm operators

Abstract

This paper consists of four parts. It begins by using the authors' generalized Schauder formula, [41], and the algebraic multiplicity, X, of Esquinas and López-Gómez [15,14,31] 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, [42]. This facilitates reformulating and refining all existing results in a compact and unifying way. Then, the local structure of the solution set of analytic nonlinearities F(Y,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. Finally, the unilateral theorems of [31,33], as well as the refinement of Shi and Wang [53], 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.