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.
We show how to correct errors in [1, § 4] caused by the incorrect inequality [1, (4.2)].Here we only point out main corrected points and refer readers to [2, § 4] for a completely rewritten version of [1, § 4]. After removing the incorrect inequality [1, (4.2)] some corrections to the arguments in [1, § 4] should be made.• The original (q * 1 ) and (q * 3 ) should be replaced by the following slightly stronger ones:(q * 1 ) There exist constants c 1 > 0, r ∈ (0, 1) and a function E ∈ L 2 (Ω) such that |q(x, t)| ≤ E(x) + c 1 |t| r for almost x ∈ Ω and for all t ∈ R. (q * 3 ) For almost every x ∈ Ω the function R t → q(x, t) is differentiable and Ω × R (x, t) → q t (x, t) := ∂q ∂t (x, t) is a Carthéodory function. There exist s ∈ (n/2, ∞), ∈ L s (Ω), and a bounded measurable h : R → R such that h(t) → ∈ R as |t| → ∞ and |q t (x, t)| ≤ (x)h(t) for almost every x ∈ Ω and for all t ∈ R.
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