# American Institute of Mathematical Sciences

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

## The splitting lemmas for nonsmooth functionals on Hilbert spaces I

 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.
Citation: Guangcun Lu. The splitting lemmas for nonsmooth functionals on Hilbert spaces I. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2939-2990. doi: 10.3934/dcds.2013.33.2939
##### References:
 [1] Adv. Nonlinear Stud., 9 (2009), 597-623.  Google Scholar [2] J. Funct. Anal., 186 (2001), 117-152. doi: 10.1006/jfan.2001.3789.  Google Scholar [3] Math. Z., 233 (2000), 655-677. doi: 10.1007/s002090050492.  Google Scholar [4] in "Handbook of Global Analysis" Elsevier Science Ltd, (2008), 41-73. doi: 10.1016/B978-044452833-9.50003-6.  Google Scholar [5] Ann. Sci. Math. Quebec, 22 (1998), 131-148.  Google Scholar [6] Topol. Methods Nonlinear Anal., 16 (2000), 279-306.  Google Scholar [7] Acad. Press, New York-London, 1977.  Google Scholar [8] C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 465-472.  Google Scholar [9] Bull. Amer. Math. Soc. (N.S), 9 (1983), 1-39. doi: 10.1090/S0273-0979-1983-15153-4.  Google Scholar [10] Univ. de Montreal, 97, 1985.  Google Scholar [11] Birkhäuser, 1993.  Google Scholar [12] Springer Monogaphs in Mathematics, Springer, 2005.  Google Scholar [13] C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 441-446.  Google Scholar [14] Adv. Nonlinear Stud., 9 (2009), 679-699.  Google Scholar [15] Wiley, New York, 1983.  Google Scholar [16] Springer, New York, 1990.  Google Scholar [17] J. Math. Anal. Appl., 196 (1995), 1050-1072. doi: 10.1006/jmaa.1995.1460.  Google Scholar [18] Gauthier-Villars, 1963.  Google Scholar [19] Proc. Amer. Math. Soc., 135 (2007), 921-927. doi: 10.1090/S0002-9939-06-08662-X.  Google Scholar [20] Topological Methods in Nonlinear Analysis, 29 (2007), 35-68.  Google Scholar [21] J. Funct. Anal., 124 (1994), 1-39. doi: 10.1006/jfan.1994.1096.  Google Scholar [22] A.I.H.P. Analyse Non linéaire, 6 (1989), 321-330.  Google Scholar [23] Topology, 8 (1969), 361-369.  Google Scholar [24] in "Nonlinear functional analysis and its applications, Part 1" (Berkeley, Calif., 1983), Proc. Symp. Pure Math., 45, Part 1, Providence, RI, (1986), 501-509.  Google Scholar [25] Springer-Verlag, 1975.  Google Scholar [26] in "Recent Developments in Optimization Theory and Nonlinear Analysis" (Jerusalem, 1995), Contemp. Math., 204, Amer. Math. Soc., Providence, RI, (1997), 139-147. doi: 10.1090/conm/204/02627.  Google Scholar [27] Nonlinear Analysis, 36 (1999), 943-960. doi: 10.1016/S0362-546X(97)00701-3.  Google Scholar [28] John wiley & Sons. Ins. 1978.  Google Scholar [29] $2^{nd}$ edition, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4684-0265-0.  Google Scholar [30] J. Funct. Anal., 221 (2005), 439-455. doi: 10.1016/j.jfa.2004.09.010.  Google Scholar [31] J. Differential Equations, 244 (2008), 2498-2528. doi: 10.1016/j.jde.2008.02.021.  Google Scholar [32] J. Funct. Anal., 261 (2011), 542-589. 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] Bull. Soc. Math., Belgique (B), 37 (1985), 23-29.  Google Scholar [37] Applied Mathematical Sciences 74, Springer-Verlag, New York, 1989.  Google Scholar [38] Abstract and Applied Analysis, 5 (2000), 113-118. doi: 10.1155/S1085337500000245.  Google Scholar [39] J. Anal. Math., 76 (1998), 289-319. doi: 10.1007/BF02786939.  Google Scholar [40] Mathematical Surveys and Monographs 161, American Mathematical Society, Providence Rhode Island 2010.  Google Scholar [41] Academic Press, New York-London, 1971.  Google Scholar [42] Springer, Berlin, 2007. doi: 10.1007/978-3-540-71333-3.  Google Scholar [43] [in Russian], Naukova Dumka, Kiev 1973.  Google Scholar [44] Teubner, Leipzig, 1986.  Google Scholar [45] Math. Ann., 263 (1983), 303-312. doi: 10.1007/BF01457133.  Google Scholar [46] J. Sov. Math., 67 (1993), 2713-2811. doi: 10.1007/BF01455151.  Google Scholar

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##### References:
 [1] Adv. Nonlinear Stud., 9 (2009), 597-623.  Google Scholar [2] J. Funct. Anal., 186 (2001), 117-152. doi: 10.1006/jfan.2001.3789.  Google Scholar [3] Math. Z., 233 (2000), 655-677. doi: 10.1007/s002090050492.  Google Scholar [4] in "Handbook of Global Analysis" Elsevier Science Ltd, (2008), 41-73. doi: 10.1016/B978-044452833-9.50003-6.  Google Scholar [5] Ann. Sci. Math. Quebec, 22 (1998), 131-148.  Google Scholar [6] Topol. Methods Nonlinear Anal., 16 (2000), 279-306.  Google Scholar [7] Acad. Press, New York-London, 1977.  Google Scholar [8] C. R. Acad. Sci. Paris Sér. I Math., 317 (1993), 465-472.  Google Scholar [9] Bull. Amer. Math. Soc. (N.S), 9 (1983), 1-39. doi: 10.1090/S0273-0979-1983-15153-4.  Google Scholar [10] Univ. de Montreal, 97, 1985.  Google Scholar [11] Birkhäuser, 1993.  Google Scholar [12] Springer Monogaphs in Mathematics, Springer, 2005.  Google Scholar [13] C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 441-446.  Google Scholar [14] Adv. Nonlinear Stud., 9 (2009), 679-699.  Google Scholar [15] Wiley, New York, 1983.  Google Scholar [16] Springer, New York, 1990.  Google Scholar [17] J. Math. Anal. Appl., 196 (1995), 1050-1072. doi: 10.1006/jmaa.1995.1460.  Google Scholar [18] Gauthier-Villars, 1963.  Google Scholar [19] Proc. Amer. Math. Soc., 135 (2007), 921-927. doi: 10.1090/S0002-9939-06-08662-X.  Google Scholar [20] Topological Methods in Nonlinear Analysis, 29 (2007), 35-68.  Google Scholar [21] J. Funct. Anal., 124 (1994), 1-39. doi: 10.1006/jfan.1994.1096.  Google Scholar [22] A.I.H.P. Analyse Non linéaire, 6 (1989), 321-330.  Google Scholar [23] Topology, 8 (1969), 361-369.  Google Scholar [24] in "Nonlinear functional analysis and its applications, Part 1" (Berkeley, Calif., 1983), Proc. Symp. Pure Math., 45, Part 1, Providence, RI, (1986), 501-509.  Google Scholar [25] Springer-Verlag, 1975.  Google Scholar [26] in "Recent Developments in Optimization Theory and Nonlinear Analysis" (Jerusalem, 1995), Contemp. Math., 204, Amer. Math. Soc., Providence, RI, (1997), 139-147. doi: 10.1090/conm/204/02627.  Google Scholar [27] Nonlinear Analysis, 36 (1999), 943-960. doi: 10.1016/S0362-546X(97)00701-3.  Google Scholar [28] John wiley & Sons. Ins. 1978.  Google Scholar [29] $2^{nd}$ edition, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4684-0265-0.  Google Scholar [30] J. Funct. Anal., 221 (2005), 439-455. doi: 10.1016/j.jfa.2004.09.010.  Google Scholar [31] J. Differential Equations, 244 (2008), 2498-2528. doi: 10.1016/j.jde.2008.02.021.  Google Scholar [32] J. Funct. Anal., 261 (2011), 542-589. 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] Bull. Soc. Math., Belgique (B), 37 (1985), 23-29.  Google Scholar [37] Applied Mathematical Sciences 74, Springer-Verlag, New York, 1989.  Google Scholar [38] Abstract and Applied Analysis, 5 (2000), 113-118. doi: 10.1155/S1085337500000245.  Google Scholar [39] J. Anal. Math., 76 (1998), 289-319. doi: 10.1007/BF02786939.  Google Scholar [40] Mathematical Surveys and Monographs 161, American Mathematical Society, Providence Rhode Island 2010.  Google Scholar [41] Academic Press, New York-London, 1971.  Google Scholar [42] Springer, Berlin, 2007. doi: 10.1007/978-3-540-71333-3.  Google Scholar [43] [in Russian], Naukova Dumka, Kiev 1973.  Google Scholar [44] Teubner, Leipzig, 1986.  Google Scholar [45] Math. Ann., 263 (1983), 303-312. doi: 10.1007/BF01457133.  Google Scholar [46] J. Sov. Math., 67 (1993), 2713-2811. doi: 10.1007/BF01455151.  Google Scholar
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