Bifurcation Theory for Fredholm Operators

TL;DR

This paper unifies and refines local and global bifurcation results for Fredholm operators using advanced algebraic tools, and explores the structure of solution sets with applications to nonlinear boundary value problems.

Contribution

It introduces a unified framework for bifurcation theory of Fredholm operators using the Fitzpatrick-Pejsachowicz-Rabier degree and connects bifurcation with algebraic geometry concepts.

Findings

Unified formulation of bifurcation results for Fredholm operators.

Analysis of solution set structure at degenerate eigenvalues.

Application to nonlinear boundary value problems involving mean curvature.

Abstract

This paper consists of four parts. It begins by using the authors's generalized Schauder formula, [50], and the algebraic multiplicity, $\chi$, of Esquinas and L\'opez-G\'omez [18,17,40] 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, [51]. 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 $\mathfrak{F}(\lambda,u)=0$ 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. Finally, the unilateral theorems of [40,42], as well as the refinement of Xi and Wang [63], 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.