Parameterized splitting theorems and bifurcations for potential operators, Part I: Abstract theory

TL;DR

This paper develops a generalized theoretical framework for bifurcation analysis of potential operators, extending classical theorems to less smooth functionals, thereby enabling new studies in quasi-linear elliptic boundary value problems.

Contribution

It introduces parameterized splitting theorems in Morse theory that weaken differentiability assumptions, broadening bifurcation analysis applicability.

Findings

Generalized bifurcation theorems for potential operators

Weakened differentiability conditions for functionals

Application potential to quasi-linear elliptic boundary value problems

Abstract

This is the first part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using parameterized versions of splitting theorems in Morse theory we generalize some famous bifurcation theorems for potential operators by weakening standard assumptions on the differentiability of the involved functionals, which opens up a way of bifurcation studies for quasi-linear elliptic boundary value problems.