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

Abstract

<p style='text-indent:20px;'>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.</p>

Bifurcations for Hamiltonian systems via dual variational principle

“…In this section we study bifurcations for solutions of the problem (1.2) with theorems in [33,Sections 3,4,5].…”

“…But such methods seem not to be effective for getting analogues of Theorems 5.5, 5.6, 5.7. We shall prove them with theories developed in [33,34], see [35]. (ii) It is natural to study bifurcations for other Hamiltonian boundary problems, see [36].…”

“…In fact, it is only C 1 and twice Gâteaux-differentiable. Fortunately, we may prove that the latter conditions are sufficient for using abstract bifurcation theorems proved by the author in [33] or their proof ideas. Each Ψ K (λ, •) has finite Morse indexes and nullities at all critical points, and these numbers may be related to some kind of Maslov indexes of corresponding solutions of (1.2).…”

“…It is well-known that both Φ λ and −Φ λ have infinite Morse indexes at critical points. Therefore our abstract bifurcation theorems in [33] cannot be directly applied to them. There are two methods to overcome this difficulty, the saddle point reduction by Amann and Zehnder and the dual variational principle by Clarke and Ekeland.…”

“…Bifurcation results for (1.7) can be derived from those of (1.2). However, Theorems 5.5, 5.6, 5.7 about bifurcations of (1.11) cannot be directly proved with theorems in [33]. We need use the proof ideas of the results in [33, §5] and other techniques to complete their proofs.…”