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On topological Morse theory

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Abstract

Starting from the concept of Morse critical point, introduced in [A. Ioffe and E. Schwartzman, J. Math. Pures Appl. (9), 75 (1996), 125–153], we propose a purely topological approach to Morse theory.

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References

  1. Benci V., Giannoni F.: Morse theory for C 1-functionals and Conley blocks. Topol. Methods Nonlinear Anal. 4, 365–398 (1994)

    MathSciNet  MATH  Google Scholar 

  2. K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems. Progr. Nonlinear Differential Equations Appl. 6, Birkhäuser Boston, Boston, MA, 1993.

  3. Cingolani S., Degiovanni M.: Nontrivial solutions for p-Laplace equations with right-hand side having p-linear growth at infinity. Comm. Partial Differential Equations 30, 1191–1203 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Corvellec J.-N.: Morse theory for continuous functionals. J. Math. Anal. Appl. 196, 1050–1072 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Corvellec J.-N.: A general approach to the min-max principle. Z. Anal. Anwend. 16, 405–433 (1997)

    MathSciNet  MATH  Google Scholar 

  6. Corvellec J.-N.: On the second deformation lemma. Topol. Methods Nonlinear Anal 17, 55–66 (2001)

    MathSciNet  MATH  Google Scholar 

  7. Corvellec J.-N., Degiovanni M.: Nontrivial solutions of quasilinear equations via nonsmooth Morse theory. J. Differential Equations 136, 268–293 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Degiovanni M.: On Morse theory for continuous functionals. Conf. Semin. Mat. Univ. Bari 290, 1–22 (2003)

    MathSciNet  Google Scholar 

  9. Degiovanni M.: On topological and metric critical point theory. J. Fixed Point Theory Appl 7, 85–102 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Degiovanni M., Lancelotti S.: Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire 24, 907–919 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Degiovanni M., Marzocchi M.: A critical point theory for nonsmooth functionals. Ann. Mat. Pura Appl. 167(4), 73–100 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fadell E.R., Rabinowitz P.H.: Bifurcation for odd potential operators and an alternative topological index. J. Funct. Anal. 26, 48–67 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fadell E.R., Rabinowitz P.H.: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math 45, 139–174 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  14. Frigon M.: On a new notion of linking and application to elliptic problems at resonance. J. Differential Equations 153, 96–120 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Girouard A.: Relative homological linking in critical point theory. Topol. Methods Nonlinear Anal 30, 211–221 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Gromoll D., Meyer W.: On differentiable functions with isolated critical points. Topology 8, 361–369 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hu S. T.: Theory of Retracts. Wayne State University Press, Detroit (1965)

    MATH  Google Scholar 

  18. Ioffe A., Schwartzman E.: Metric critical point theory. I. Morse regularity and homotopic stability of a minimum. J. Math. Pures Appl. 75(9), 125–153 (1996)

    MathSciNet  MATH  Google Scholar 

  19. Katriel G.: Mountain pass theorems and global homeomorphism theorems. Ann. Inst. H. Poincaré Anal. Non Linéaire 11, 189–209 (1994)

    MathSciNet  MATH  Google Scholar 

  20. Kelley J.L.: General Topology. Springer-Verlag, New York (1975)

    MATH  Google Scholar 

  21. Lancelotti S.: Morse index estimates for continuous functionals associated with quasilinear elliptic equations. Adv. Differential Equations 7, 99–128 (2002)

    MathSciNet  MATH  Google Scholar 

  22. Marino A., Micheletti A.M., Pistoia A.: A nonsymmetric asymptotically linear elliptic problem. Topol. Methods Nonlinear Anal. 4, 289–339 (1994)

    MathSciNet  MATH  Google Scholar 

  23. Marino A., Prodi G.: Metodi perturbativi nella teoria di Morse. Boll. Un. Mat. Ital 4(11), 1–32 (1975)

    MathSciNet  Google Scholar 

  24. A. Masiello, Variational Methods in Lorentzian Geometry. Pitman Research Notes in Mathematics Series 309, Longman Scientific & Technical, Harlow, 1994.

  25. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems. Appl. Math. Sci. 74, Springer-Verlag, New York, 1989.

  26. Morse M.: Functional topology and abstract variational theory. Proc. Natl. Acad. Sci. USA 22, 313–319 (1936)

    Article  Google Scholar 

  27. Morse M.: Functional topology and abstract variational theory. Ann. of Math. 38(2), 386–449 (1937)

    Article  MathSciNet  Google Scholar 

  28. Morse M., Cairns S.S.: Critical Point Theory in Global Analysis and Differential Topology: An Introduction Pure and Applied Mathematics 33. Academic Press, New York (1969)

    MATH  Google Scholar 

  29. O’Neill B.: Semi-Riemannian Geometry Pure and Applied Mathematics 103. Academic Press, New York (1983)

    Google Scholar 

  30. Palais R.S.: Morse theory on Hilbert manifolds. Topology 2, 299–340 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  31. Palais R.S.: Homotopy theory of infinite dimensional manifolds. Topology 5, 1–16 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  32. R. S. Palais, Critical point theory and the minimax principle. In: Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif, 1968), Amer. Math. Soc., Providence, RI, 1970, 185–212.

  33. Palais R.S., Smale S.: A generalized Morse theory. Bull. Amer. Math. Soc 70, 165–172 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  34. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics 65, Amer. Math. Soc., Providence, RI, 1986.

  35. Shafrir I.: A deformation lemma. C. R. Acad. Sci. Paris Sér. I Math. 313, 599–602 (1991)

    MathSciNet  MATH  Google Scholar 

  36. Smale S.: Morse theory and a non-linear generalization of the Dirichlet problem. Ann. of Math 2(80), 382–396 (1964)

    Article  MathSciNet  Google Scholar 

  37. Spanier E.H.: Algebraic Topology. McGraw–Hill, New York (1966)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei, 41, 25121, Brescia, Italy

    Marco Degiovanni

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  1. Marco Degiovanni
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Correspondence to Marco Degiovanni.

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To Richard S. Palais

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Degiovanni, M. On topological Morse theory. J. Fixed Point Theory Appl. 10, 197–218 (2011). https://doi.org/10.1007/s11784-011-0066-8

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