Morse theory methods for a class of quasi-linear elliptic systems of higher order

Abstract

We develop the local Morse theory for a class of non-twice continuously differentiable functionals on Hilbert spaces, including a new generalization of the Gromoll-Meyer's splitting theorem and a weaker Marino-Prodi perturbation type result. With them some critical point theorems and famous bifurcation theorems are generalized. Then we show that these are applicable to studies of quasi-linear elliptic equations and systems of higher order given by multi-dimensional variational problems as in (1.3).

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

“…For example, suppose that the eigenvalue λ * is also isolated. Then for each λ = λ * close to λ * the origin 0 ∈ H is a nondegenerate critical point of L λ in the sense of [45], in particular an isolated critical point of L λ and thus 0 ∈ H • is such a critical point of L • λ as well. Under some additional conditions we can use the parameterized shifting theorem in [45] to compute critical groups of L • λ at 0 ∈ H 0 and conclude that L • λ takes a local maximum (resp.…”

“…Thus the Rabinowitz bifurcation theorem are inapplicable for studying bifurcations of quasilinear elliptic equations of potential type since the corresponding variational functionals on natural chosen Sobolev spaces cannot be of class C 2 in general. Recently, the author developed Morse theory methods for a class of quasilinear elliptic equations by proving some splitting theorems for some classes of non-C 2 functionals [40]- [45]. In their proofs our finite-dimensional reductions either required weaker differentiability for potential operators or were completed on a smaller space.…”

“…Our main aim is to study generalizations of some previous famous bifurcation theorems such as those by Krasnosel'skii [37], by Rabinowitz [59], and by Fadelll and Rabinowitz [26,27] with parameterized versions of two new splitting theorems established by the author [45] and [40,41,43] (see Appendix A) and the parameterized version of a splitting lemma by Bobylev and Burman [8] (see Appendix B). The following are our basic assumptions related to them.…”

“…By [45,Lemma 2.7] the condition (D4) is equivalent to the following (D4*) There exist positive constants η 0 > 0 and C 0 > 0 such that BH (0, η 0 ) ⊂ U and (P (x)u, u) ≥ C 0 u 2 ∀u ∈ H, ∀x ∈ BH (0, η 0 ) ∩ X.…”

“…Some comparisons of our results with previous work are also given in Section 3.3. In Sections 4, 5 we shall use the parameterized versions of splitting theorems [45,40,41,43] to generalize some older bifurcation theorems. Section 4 is dedicated to some generalizations of Rabinowitz bifurcation theorem [59].…”