The splitting lemmas for nonsmooth functional on Hilbert spaces II. The case at infinity.

Abstract

We generalize the Bartsch-Li's splitting lemma at infinity for C 2 -functionals in [2] and some later variants of it to a class of continuously directional differentiable functionals on Hilbert spaces. Different from the previous flow methods our proof is to combine the ideas of the Morse-Palais lemma due to Duc-Hung-Khai [9] with some techniques from [11], [17], [18]. A simple application is also presented.

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

“…However, it is difficult to compute critical groups for non-twice continuously differentiable functionals. In this section, with helps of splitting theorems in [41,42,45] we give some general bifurcation results for potential operator families of a class of non-twice continuously differentiable functionals.…”

“…Once parameterized versions of other splitting theorems on Banach spaces are established, our above methods are applicable. For example, we may also write the parameterized versions of the splitting lemmas at infinity in [42], and then use them to derive some theorems of bifurcations at infinity. These will be given otherwise.…”

Corrigendum to "The splitting lemmas for nonsmooth functionals on Hilbert spaces II. The case at infinity" (Topol. Methods Nonlinear Anal. 44 (2014), 277-335)

“…• Two lines above Proposition 4.2 of [1] should be changed into: Since |q t (x, t)| ≤ (x)h(t) by (q * 3 ), 1/s + 1/η(s, n) + 2/ξ(s, n) = 1, η(s, n) > 1, and 2s/(s − 1) < ξ(s, n) < 2n/(n − 2) for n > 2, using the generalized Hölder inequality and Sobolev embedding theorem, we deduce…”

“…• (b) of [1,Proposition 4.2] should be replaced by (b) Under the assumption (q * 3 ), J is C 2 and J (u) := D(∇J)(u) = B(u) for all u ∈ H. Moreover, if a = λ m it holds with the constant c(s, n, Ω) in (0.4) that…”

“…• The last two lines on [1,325] (or the equalities [1, (4.13)]) should be removed. And [1,Claim 4.4] should be replaced by: Claim 4.4. For given numbers ρ > 0 and ε > 0 there exists…”