### 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 [T. Bartsch and S.-J. Li, < i> Critical point theory

for asymptotically quadratic functionals and applications to problems with resonance< /i> ,

Nonlinear Anal. < b> 28< /b> (1997), no. 3, 419-441]

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 [D.M. Duc, T.V. Hung and N.T. Khai, < i> Morse-Palais lemma

for nonsmooth

functionals on normed spaces< /i> , Proc. Amer. Math. Soc. < b> 135< /b> (2007),

no. 3., 921-927] with some techniques from [M. Jiang, < i> A generalization

of Morse lemma and its applications< /i> , Nonlinear Anal. < b> 36< /b>

(1999), 943-960],

[I.V. Skrypnik, < i> Nonlinear Elliptic Equations of a Higher Order< /i> , Naukova Dumka,

Kiev (1973)], [S.A. Vakhrameev, < i> Critical point theory for smooth functions

on Hilbert manifolds with singularities and its application to some optimal

control problems< /i> , J. Sov. Math. < b> 67< /b> (1993), no. 1, 2713-2811].

A simple application is also presented.

infinity for $C^2$-functionals in [T. Bartsch and S.-J. Li, < i> Critical point theory

for asymptotically quadratic functionals and applications to problems with resonance< /i> ,

Nonlinear Anal. < b> 28< /b> (1997), no. 3, 419-441]

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 [D.M. Duc, T.V. Hung and N.T. Khai, < i> Morse-Palais lemma

for nonsmooth

functionals on normed spaces< /i> , Proc. Amer. Math. Soc. < b> 135< /b> (2007),

no. 3., 921-927] with some techniques from [M. Jiang, < i> A generalization

of Morse lemma and its applications< /i> , Nonlinear Anal. < b> 36< /b>

(1999), 943-960],

[I.V. Skrypnik, < i> Nonlinear Elliptic Equations of a Higher Order< /i> , Naukova Dumka,

Kiev (1973)], [S.A. Vakhrameev, < i> Critical point theory for smooth functions

on Hilbert manifolds with singularities and its application to some optimal

control problems< /i> , J. Sov. Math. < b> 67< /b> (1993), no. 1, 2713-2811].

A simple application is also presented.

#### Keywords

Nonsmooth functional; splitting lemma at infinity; elliptic boundary value problems

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