American Institute of Mathematical Sciences

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July  2013, 33(7): 2939-2990. doi: 10.3934/dcds.2013.33.2939

The splitting lemmas for nonsmooth functionals on Hilbert spaces I

Guangcun Lu 1,

1. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Received  August 2011 Revised  October 2012 Published  January 2013

The Gromoll-Meyer's generalized Morse lemma (so called splitting lemma) near degenerate critical points on Hilbert spaces, which is one of key results in infinite dimensional Morse theory, is usually stated for at least $C^2$-smooth functionals. It obstructs one using Morse theory to study most of variational problems of form $F(u)=\int_\Omega f(x, u,\cdots, D^mu)dx$ as in (1.1). In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower than $C^1$) on a Hilbert space $H$ which have higher smoothness (but lower than $C^2$) on a densely and continuously imbedded Banach space $X\subset H$ near a critical point lying in $X$. (This splitting theorem generalize almost all previous ones to my knowledge). Moreover, a new theorem of Poincaré-Hopf type and a relation between critical groups of the functional on $H$ and $X$ are given. Different from the usual implicit function theorem method and dynamical system one our proof is to combine the ideas of the Morse-Palais lemma due to Duc-Hung-Khai [19] with some techniques from [27,43,46]. Our theory is applicable to the Lagrangian systems on compact manifolds and boundary value problems for a large class of nonlinear higher order elliptic equations.
Keywords: splitting lemma, Critical groups, shifting theorem..
Mathematics Subject Classification: Primary: 58E05, 49J52, 49J4.
Citation: Guangcun Lu. The splitting lemmas for nonsmooth functionals on Hilbert spaces I. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2939-2990. doi: 10.3934/dcds.2013.33.2939
References:
[1]

A. Abbondandolo and M. Schwarz, A smooth pseudo-gradient for the Lagrangian action functional,, Adv. Nonlinear Stud., 9 (2009), 597.   Google Scholar

[2]

T. Bartsch, Critical point theory on partially ordered hilbert spaces,, J. Funct. Anal., 186 (2001), 117.  doi: 10.1006/jfan.2001.3789.  Google Scholar

[3]

T. Bartsch, K.-C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems,, Math. Z., 233 (2000), 655.  doi: 10.1007/s002090050492.  Google Scholar

[4]

T. Bartsch, A. Szulkin and M. Willem, Morse theory and nonlinear differential equations,, in, (2008), 41.  doi: 10.1016/B978-044452833-9.50003-6.  Google Scholar

[5]

P. Benevieri and M. Furi, A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree,, Ann. Sci. Math. Quebec, 22 (1998), 131.   Google Scholar

[6]

P. Benevieri and M. Furi, On the concept of orientability for Fredholm maps between real Banach manifolds,, Topol. Methods Nonlinear Anal., 16 (2000), 279.   Google Scholar

[7]

M. Berger, "Nonlinearity and Functional Analysis,", Acad. Press, (1977).   Google Scholar

[8]

H. Brézis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers,, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 465.   Google Scholar

[9]

F. E. Browder, Fixed point theory and nonlinear problem,, Bull. Amer. Math. Soc. (N.S), 9 (1983), 1.  doi: 10.1090/S0273-0979-1983-15153-4.  Google Scholar

[10]

K. C. Chang, "Infinite Dimensional Morse Theory and its Applications,", Univ. de Montreal, 97 (1985).   Google Scholar

[11]

K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problem,", Birkhäuser, (1993).   Google Scholar

[12]

K. C. Chang, "Methods in Nonlinear Analysis,", Springer Monogaphs in Mathematics, (2005).   Google Scholar

[13]

K. C. Chang, $H^1$ versus $C^1$ isolated critical points,, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 441.   Google Scholar

[14]

S. Cingolani and M. Degiovanni, On the Poincaré-Hopf theorem for functionals defined on Banach spaces,, Adv. Nonlinear Stud., 9 (2009), 679.   Google Scholar

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F. H. Clarke, "Optimization and Nonsmooth Analysis,", Wiley, (1983).   Google Scholar

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J. N. Corvellec, Morse theory for continuous functionals,, J. Math. Anal. Appl., 196 (1995), 1050.  doi: 10.1006/jmaa.1995.1460.  Google Scholar

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J. Dieudonné, "Fondements de L'Analyse Moderne,", Gauthier-Villars, (1963).   Google Scholar

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D. M. Duc, T. V. Hung and N. T. Khai, Morse-Palais lemma for nonsmooth functionals on normed spaces,, Proc. Amer. Math. Soc., 135 (2007), 921.  doi: 10.1090/S0002-9939-06-08662-X.  Google Scholar

[20]

D. M. Duc, T. V. Hung and N. T. Khai, Critical points of non-$C^2$ functionals,, Topological Methods in Nonlinear Analysis, 29 (2007), 35.   Google Scholar

[21]

P. M. Fitzpatrick, J. Pejsachowicz and P. J. Rabier, Orientability of fredholm families and topological degree for orientable nonlinear fredholm mappings,, J. Funct. Anal., 124 (1994), 1.  doi: 10.1006/jfan.1994.1096.  Google Scholar

[22]

N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points,, A.I.H.P. Analyse Non linéaire, 6 (1989), 321.   Google Scholar

[23]

D. Gromoll and W. Meyer, On differentiable functions with isolated critical points,, Topology, 8 (1969), 361.   Google Scholar

[24]

H. Hofer, The topological degree at a critical point of mountain pass type,, in, 45 (1986), 501.   Google Scholar

[25]

D. Husemoller, "Fibre Bundle,", Springer-Verlag, (1975).   Google Scholar

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A. Ioffe and E. Schwartzman, Parametric Morse lemmas for $C^{1,1}$-functions,, in, 204 (1997), 139.  doi: 10.1090/conm/204/02627.  Google Scholar

[27]

M. Jiang, A generalization of Morse lemma and its applications,, Nonlinear Analysis, 36 (1999), 943.  doi: 10.1016/S0362-546X(97)00701-3.  Google Scholar

[28]

E. Kreyszig, "Introduction Functional Analysis with Applications,", John wiley & Sons. Ins. 1978., (1978).   Google Scholar

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S. Lang, "Differential Manifolds,", $2^{nd}$ edition, (1985).  doi: 10.1007/978-1-4684-0265-0.  Google Scholar

[30]

C. Li, S. -J. Li and J. Liu, Splitting theorem, Poincare-Hopf theorem and jumping nonlinear problems,, J. Funct. Anal., 221 (2005), 439.  doi: 10.1016/j.jfa.2004.09.010.  Google Scholar

[31]

C. Li, S.-J. Li, Z. Liu and J. Pan, On the Fucík spectrum,, J. Differential Equations, 244 (2008), 2498.  doi: 10.1016/j.jde.2008.02.021.  Google Scholar

[32]

G. Lu, Corrigendum to "The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems" [J. Funct. Anal. 256(9)(2009)2967-3034],, J. Funct. Anal., 261 (2011), 542.  doi: 10.1016/j.jfa.2009.01.001.  Google Scholar

[33]

G. Lu, The splitting lemmas for nonsmooth functionals on Hilbert spaces,, preprint, ().   Google Scholar

[34]

G. Lu, Some critical point theorems and applications,, preprint, ().   Google Scholar

[35]

G. Lu, Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds,, preprint, ().   Google Scholar

[36]

Jean Mawhin and Michel Willem, On the generalized Morse Lemma,, Bull. Soc. Math., 37 (1985), 23.   Google Scholar

[37]

Jean Mawhin and Michel Willem, "Critical Point Theory and Hamiltonian Systems,", Applied Mathematical Sciences 74, 74 (1989).   Google Scholar

[38]

A. A. Moura and F. M. de Souza, A Morse lemma for degenerate critical points with low differentiability,, Abstract and Applied Analysis, 5 (2000), 113.  doi: 10.1155/S1085337500000245.  Google Scholar

[39]

J. Pejsachowicz and P. R. Rabier, Degree theory for $C^1$ Fredholm mappings of index $0$,, J. Anal. Math., 76 (1998), 289.  doi: 10.1007/BF02786939.  Google Scholar

[40]

K. Perera, R. P. Agarwal and Donal O'Regan, "Morse Theoretic Aspects of $p$-Laplacian Type Operators,", Mathematical Surveys and Monographs 161, 161 (2010).   Google Scholar

[41]

M. Schechter, "Principles of Functional Analysis,", Academic Press, (1971).   Google Scholar

[42]

W. Schirotzek, "Nonsmooth Analysis,", Springer, (2007).  doi: 10.1007/978-3-540-71333-3.  Google Scholar

[43]

I. V. Skrypnik, "Nonlinear Elliptic Equations of a Higher Order,", [in Russian], (1973).   Google Scholar

[44]

I. V. Skrypnik, "Nonlinear Elliptic Boundary Value Problems,", Teubner, (1986).   Google Scholar

[45]

A. Tromba, A sufficient condition for a critical point of a functional to be a minimum and its application to Plateau's problem,, Math. Ann., 263 (1983), 303.  doi: 10.1007/BF01457133.  Google Scholar

[46]

S. A. Vakhrameev, Critical point theory for smooth functions on Hilbert manifolds with singularities and its application to some optimal control problems,, J. Sov. Math., 67 (1993), 2713.  doi: 10.1007/BF01455151.  Google Scholar

show all references

References:
[1]

A. Abbondandolo and M. Schwarz, A smooth pseudo-gradient for the Lagrangian action functional,, Adv. Nonlinear Stud., 9 (2009), 597.   Google Scholar

[2]

T. Bartsch, Critical point theory on partially ordered hilbert spaces,, J. Funct. Anal., 186 (2001), 117.  doi: 10.1006/jfan.2001.3789.  Google Scholar

[3]

T. Bartsch, K.-C. Chang and Z.-Q. Wang, On the Morse indices of sign changing solutions of nonlinear elliptic problems,, Math. Z., 233 (2000), 655.  doi: 10.1007/s002090050492.  Google Scholar

[4]

T. Bartsch, A. Szulkin and M. Willem, Morse theory and nonlinear differential equations,, in, (2008), 41.  doi: 10.1016/B978-044452833-9.50003-6.  Google Scholar

[5]

P. Benevieri and M. Furi, A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree,, Ann. Sci. Math. Quebec, 22 (1998), 131.   Google Scholar

[6]

P. Benevieri and M. Furi, On the concept of orientability for Fredholm maps between real Banach manifolds,, Topol. Methods Nonlinear Anal., 16 (2000), 279.   Google Scholar

[7]

M. Berger, "Nonlinearity and Functional Analysis,", Acad. Press, (1977).   Google Scholar

[8]

H. Brézis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers,, C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 465.   Google Scholar

[9]

F. E. Browder, Fixed point theory and nonlinear problem,, Bull. Amer. Math. Soc. (N.S), 9 (1983), 1.  doi: 10.1090/S0273-0979-1983-15153-4.  Google Scholar

[10]

K. C. Chang, "Infinite Dimensional Morse Theory and its Applications,", Univ. de Montreal, 97 (1985).   Google Scholar

[11]

K. C. Chang, "Infinite Dimensional Morse Theory and Multiple Solution Problem,", Birkhäuser, (1993).   Google Scholar

[12]

K. C. Chang, "Methods in Nonlinear Analysis,", Springer Monogaphs in Mathematics, (2005).   Google Scholar

[13]

K. C. Chang, $H^1$ versus $C^1$ isolated critical points,, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 441.   Google Scholar

[14]

S. Cingolani and M. Degiovanni, On the Poincaré-Hopf theorem for functionals defined on Banach spaces,, Adv. Nonlinear Stud., 9 (2009), 679.   Google Scholar

[15]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", Wiley, (1983).   Google Scholar

[16]

J. B. Conway, "A Course in Functional Analysis,", Springer, (1990).   Google Scholar

[17]

J. N. Corvellec, Morse theory for continuous functionals,, J. Math. Anal. Appl., 196 (1995), 1050.  doi: 10.1006/jmaa.1995.1460.  Google Scholar

[18]

J. Dieudonné, "Fondements de L'Analyse Moderne,", Gauthier-Villars, (1963).   Google Scholar

[19]

D. M. Duc, T. V. Hung and N. T. Khai, Morse-Palais lemma for nonsmooth functionals on normed spaces,, Proc. Amer. Math. Soc., 135 (2007), 921.  doi: 10.1090/S0002-9939-06-08662-X.  Google Scholar

[20]

D. M. Duc, T. V. Hung and N. T. Khai, Critical points of non-$C^2$ functionals,, Topological Methods in Nonlinear Analysis, 29 (2007), 35.   Google Scholar

[21]

P. M. Fitzpatrick, J. Pejsachowicz and P. J. Rabier, Orientability of fredholm families and topological degree for orientable nonlinear fredholm mappings,, J. Funct. Anal., 124 (1994), 1.  doi: 10.1006/jfan.1994.1096.  Google Scholar

[22]

N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points,, A.I.H.P. Analyse Non linéaire, 6 (1989), 321.   Google Scholar

[23]

D. Gromoll and W. Meyer, On differentiable functions with isolated critical points,, Topology, 8 (1969), 361.   Google Scholar

[24]

H. Hofer, The topological degree at a critical point of mountain pass type,, in, 45 (1986), 501.   Google Scholar

[25]

D. Husemoller, "Fibre Bundle,", Springer-Verlag, (1975).   Google Scholar

[26]

A. Ioffe and E. Schwartzman, Parametric Morse lemmas for $C^{1,1}$-functions,, in, 204 (1997), 139.  doi: 10.1090/conm/204/02627.  Google Scholar

[27]

M. Jiang, A generalization of Morse lemma and its applications,, Nonlinear Analysis, 36 (1999), 943.  doi: 10.1016/S0362-546X(97)00701-3.  Google Scholar

[28]

E. Kreyszig, "Introduction Functional Analysis with Applications,", John wiley & Sons. Ins. 1978., (1978).   Google Scholar

[29]

S. Lang, "Differential Manifolds,", $2^{nd}$ edition, (1985).  doi: 10.1007/978-1-4684-0265-0.  Google Scholar

[30]

C. Li, S. -J. Li and J. Liu, Splitting theorem, Poincare-Hopf theorem and jumping nonlinear problems,, J. Funct. Anal., 221 (2005), 439.  doi: 10.1016/j.jfa.2004.09.010.  Google Scholar

[31]

C. Li, S.-J. Li, Z. Liu and J. Pan, On the Fucík spectrum,, J. Differential Equations, 244 (2008), 2498.  doi: 10.1016/j.jde.2008.02.021.  Google Scholar

[32]

G. Lu, Corrigendum to "The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems" [J. Funct. Anal. 256(9)(2009)2967-3034],, J. Funct. Anal., 261 (2011), 542.  doi: 10.1016/j.jfa.2009.01.001.  Google Scholar

[33]

G. Lu, The splitting lemmas for nonsmooth functionals on Hilbert spaces,, preprint, ().   Google Scholar

[34]

G. Lu, Some critical point theorems and applications,, preprint, ().   Google Scholar

[35]

G. Lu, Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds,, preprint, ().   Google Scholar

[36]

Jean Mawhin and Michel Willem, On the generalized Morse Lemma,, Bull. Soc. Math., 37 (1985), 23.   Google Scholar

[37]

Jean Mawhin and Michel Willem, "Critical Point Theory and Hamiltonian Systems,", Applied Mathematical Sciences 74, 74 (1989).   Google Scholar

[38]

A. A. Moura and F. M. de Souza, A Morse lemma for degenerate critical points with low differentiability,, Abstract and Applied Analysis, 5 (2000), 113.  doi: 10.1155/S1085337500000245.  Google Scholar

[39]

J. Pejsachowicz and P. R. Rabier, Degree theory for $C^1$ Fredholm mappings of index $0$,, J. Anal. Math., 76 (1998), 289.  doi: 10.1007/BF02786939.  Google Scholar

[40]

K. Perera, R. P. Agarwal and Donal O'Regan, "Morse Theoretic Aspects of $p$-Laplacian Type Operators,", Mathematical Surveys and Monographs 161, 161 (2010).   Google Scholar

[41]

M. Schechter, "Principles of Functional Analysis,", Academic Press, (1971).   Google Scholar

[42]

W. Schirotzek, "Nonsmooth Analysis,", Springer, (2007).  doi: 10.1007/978-3-540-71333-3.  Google Scholar

[43]

I. V. Skrypnik, "Nonlinear Elliptic Equations of a Higher Order,", [in Russian], (1973).   Google Scholar

[44]

I. V. Skrypnik, "Nonlinear Elliptic Boundary Value Problems,", Teubner, (1986).   Google Scholar

[45]

A. Tromba, A sufficient condition for a critical point of a functional to be a minimum and its application to Plateau's problem,, Math. Ann., 263 (1983), 303.  doi: 10.1007/BF01457133.  Google Scholar

[46]

S. A. Vakhrameev, Critical point theory for smooth functions on Hilbert manifolds with singularities and its application to some optimal control problems,, J. Sov. Math., 67 (1993), 2713.  doi: 10.1007/BF01455151.  Google Scholar

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