Parameterized splitting theorems and bifurcations for potential operators, Part II: Applications to quasi-linear elliptic equations and systems

Abstract

<p style='text-indent:20px;'>This is the second part of a series devoting to the generalizations and applications of common theorems in variational bifurcation theory. Using abstract theorems in the first part we obtain many new bifurcation results for quasi-linear elliptic boundary value problems of higher order.</p>

Bifurcations for Hamiltonian systems via dual variational principle

“…is constant and H 0 = 0, Proposition 2.5 gives stronger results than(3.3) and(3.4). As a consequence of Theorem 3.4 we can also obtain a corresponding result to[34, Theorem 5.6].For a real τ > 0 let H : [0, τ ] × R 2n → R be a continuous function such that each H(t, •) : R 2n → R, t ∈ [0, τ ], is C2 and all its partial derivatives depend continuously on (t, z) ∈ [0, τ ] × R 2n . Let M ∈ Sp(2n, R), u 0 ∈ Ker(M − I 2n ) satisfy dH t (u 0 ) = 0 for all t ∈ [0, τ ], and let B(t) = H ′′ t (u 0 ) and ξ 2n (t) = I 2n for t ∈ [0, τ ].…”

“…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].…”

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

“…Moreover, the condition (I.d) can be replaced by Then (λ * , 0) ∈ R × U X is a bifurcation point of the equation Based on the arguments in [40], we may use Theorem 3.5 to get a generalization of [24, Theorem 5.4.1] immediately. See [46,Theorem 3.10] for a high dimensional analogue (corresponding to [24,Theorems 5.4.2 and 5.7.4]).…”