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Morse theory methods for a class of quasi-linear elliptic systems of higher order

  • Guangcun LuEmail author
Article
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Abstract

We develop the local Morse theory for a class of non-twice continuously differentiable functionals on Hilbert spaces, including a new generalization of the Gromoll–Meyer’s splitting theorem and a weaker Marino–Prodi perturbation type result. They are applicable to a wide range of multiple integrals with quasi-linear elliptic Euler equations and systems of higher order.

Mathematics Subject Classification

Primary 58E05 49J52 49J45 

Notes

Acknowledgements

The author is grateful to the anonymous referees for useful remarks.

References

  1. 1.
    Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)zbMATHGoogle Scholar
  2. 2.
    Bartsch, T., Szulkin, A., Willem, M.: Morse theory and nonlinear differential equations. In: Krupka, D., Saunders, D. (eds.) Handbook of Global Analysis, pp. 41–73. Elsevier, Amsterdam (2008)CrossRefGoogle Scholar
  3. 3.
    Berger, M.: Nonlinearity and Functional Analysis. Academic Press, New York (1977)zbMATHGoogle Scholar
  4. 4.
    Bobylev, N.A., Burman, Y.M.: Morse lemmas for multi-dimensional variational problems. Nonlinear Anal. 18, 595–604 (1992)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bott, R.: Nondegenerate critical manifold. Ann. Math. 60, 248–261 (1954)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011)zbMATHGoogle Scholar
  7. 7.
    Browder, F.E.: Nonlinear elliptic boundary value problems and the generalized topological degree. Bull. Am. Math. Soc. 76, 999–1005 (1970)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Browder, F.E.: Fixed point theory and nonlinear problem. Bull. Am. Math. Soc. (New Ser.) 9, 1–39 (1983)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Caklovic, L., Li, S.J., Willem, M.: A note on Palais-Smale condition and coercivity. Differ. Integral Equ. 3, 799–800 (1990)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Carmona, J., Cingolani, S., Martínez-Aparicio, P.-J., Vannella, G.: Regularity and Morse index of the solutions to critical quasilinear elliptic systems. Commun. Partial Differ. Equ. 38(10), 1675–1711 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chang, K.C.: Morse theory on Banach space and its applications to partial differential equations. Chin. Ann. Math. Ser. B 4, 381–399 (1983)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chang, K.C.: Infinite Dimensional Morse Theory and Multiple Solution Problem. Birkhäuser, Basel (1993)CrossRefGoogle Scholar
  13. 13.
    Chang, K.C.: Methods in Nonlinear Analysis. Springer Monogaphs in Mathematics. Springer, Berlin (2005)zbMATHGoogle Scholar
  14. 14.
    Chang, K.C., Ghoussoub, H.: The Conley index and the critical groups via an extension of Gromoll–Meyer theory. Topol. Methods Nonlinear Anal. 7, 77–93 (1996)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Chen, C.Y., Kristensen, J.: On coercive variational integrals. Nonlinear Anal. Theory Methods Appl. 153, 213–229 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cingolani, S., Degiovanni, M.: On the Poincaré–Hopf theorem for functionals defined on Banach spaces. Adv. Nonlinear Stud. 9, 679–699 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Cingolani, S., Degiovanni, M., Vannella, G.: Critical group estimates for nonregular critical points of functionals associated with quasilinear elliptic equations. J. Elliptic Parabol. Equ. 1, 75–87 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Cingolani, S., Degiovanni, M., Vannella, G.: Amann–Zehnder type results for \(p\)-Laplace problems. Ann. Mat. Pura Appl. (4) 197(2), 605–640 (2018)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Cingolani, S., Vannella, G.: Marino–Prodi perturbation type results and Morse indices of minimax critical points for a class of functionals in Banach spaces. Ann. Mat. 186, 155–183 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Dalbono, F., Portaluri, A.: Morse-Smale index theorems for elliptic boundary deformation problems. J. Differ. Equ. 253(2), 463–480 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Degiovanni, M.: On topological and metric critical point theory. J. Fixed Point Theory Appl. 7(1), 85–102 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Duc, D.M., Hung, T.V., Khai, N.T.: Morse–Palais lemma for nonsmooth functionals on normed spaces. Proc. Am. Math. Soc. 135, 921–927 (2007)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Duc, D.M., Hung, T.V., Khai, N.T.: Critical points of non-\(C^2\) functionals. Topol. Methods Nonlinear Anal. 29, 35–68 (2007)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Ekeland, I.: An inverse function theorem in Frechet spaces. Ann. Inst. Henri Poincaré Anal. Non Linéaire 28(1), 91–105 (2011)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Feckan, M.: An inverse function theorem for continuous mappings. J. Math. Anal. Appl. 185(1), 118–128 (1994)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ghoussoub, N.: Duality and Perturbation Methods in Critical Point Theory. Cambridge University Press, Cambridge (2008)zbMATHGoogle Scholar
  27. 27.
    Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Annals of Mathematics Studies. Princeton University Press, Princeton (1983)zbMATHGoogle Scholar
  28. 28.
    Gromoll, D., Meyer, W.: On differentiable functions with isolated critical points. Topology 8, 361–369 (1969)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Jiang, M.: A generalization of Morse lemma and its applications. Nonlinear Anal. 36, 943–960 (1999)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Krasnosel’skii, M.A.: Topological Methods in the Theory of Nonlinear Integral Equations. McMillan, New York (1964)zbMATHGoogle Scholar
  31. 31.
    Lazer, A., Solimini, S.: Nontrivial solutions of operator equations and Morse indices of critical points of min-max type. Nonlinear Anal. TMA 12, 761–775 (1988)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Lu, G.: Corrigendum to “The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems” [J. Funct. Anal. 256(9):2967–3034 (2009)]. J. Funct. Anal. 261, 542–589 (2011)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Lu, G.: The splitting lemmas for nonsmooth functionals on Hilbert spaces I. Discret. Contin. Dyn. Syst. 33(7), 2939–2990 (2013)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Lu, G.: The splitting lemmas for nonsmooth functionals on Hilbert spaces II. Topol. Methods Nonlinear Anal. 44, 277–335 (2014)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Lu, G.: The splitting lemmas for nonsmooth functionals on Hilbert spaces. arxiv:1102.2062
  36. 36.
    Lu, G.: Splitting lemmas for the Finsler energy functional on the space of \(H^1\)-curves. Proc. Lond. Math. Soc. 113(3), 24–76 (2016)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Lu, G.: Nonsmooth generalization of some critical point theorems for \(C^2\) functionals. Sci. Sin. Math. 46, 615–638 (2016).  https://doi.org/10.1360/N012015-00375. (in Chinese) CrossRefGoogle Scholar
  38. 38.
    Lu, G.: Morse theory methods for quasi-linear elliptic systems of higher order. arXiv:1702.06667
  39. 39.
    Lu, G.: Parameterized splitting theorems and bifurcations for potential operators. arXiv:1712.03479
  40. 40.
    Lu, G.: Variational methods for Lagrangian systems of higher order, A book in progressGoogle Scholar
  41. 41.
    Marino, A., Prodi, G.: Metodi perturbativi nella teoria di Morse. Boll. Un. Mat. Ital. 11, 1–32 (1975)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Applied Mathematical Sciences, vol. 74. Springer, New York (1989)CrossRefGoogle Scholar
  43. 43.
    Milnor, J.: Morse Theory. Annals of Mathematical Studies, vol. 51. Princeton University Press, Princeton, NJ (1963)Google Scholar
  44. 44.
    Morrey Jr., C.B.: Multiple Integrals in the Calculus of Variations. Reprint of the 1966 Classics in Mathematics. Springer, Berlin (2008)CrossRefGoogle Scholar
  45. 45.
    Morse, M.: The Calculus of Variations in the Large, vol. 18. American Mathematical Society, Colloquium Publications, Ann Arbor (1934)zbMATHGoogle Scholar
  46. 46.
    Moser, J.: Minimal solutions of variational problems on a torus. Ann. Inst. Henri Poincaré 3, 229–272 (1986)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Motreanu, D., Motreanu, V., Papageorgiou, N.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)CrossRefGoogle Scholar
  48. 48.
    Palais, R.: Morse theory on Hilbert manifolds. Topology 2, 299–340 (1963)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Palais, R.: Foundations of Global Non-linear Analysis, vol. 44. W. A. Benjamin, New York (1968)zbMATHGoogle Scholar
  50. 50.
    Palais, R.S., Smale, S.: A generalized Morse theory. Bull. Am. Math. Soc. 70, 165–172 (1964)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Perera, K., Agarwal, R.P., O’Regan, D.: Morse Theoretic Aspects of \(p\)-Laplacian Type Operators. Mathematical Surveys and Monographs, vol. 161. American Mathematical Society, Providence (2010)CrossRefGoogle Scholar
  52. 52.
    Skrypnik, I.V.: Nonlinear Elliptic Equations of a Higher Order. Naukova Dumka, Kiev (1973). (in Russian) zbMATHGoogle Scholar
  53. 53.
    Skrypnik, I.V.: Solvability and properties of solutions of nonlinear elliptic equations. J. Sov. Math. 12, 555–629 (1979)CrossRefGoogle Scholar
  54. 54.
    Skrypnik, I.V.: Methods for Analysis of Nonlinear Elliptic Boundary Value Problems. Translations of Mathematical Monographs, vol. 139. American Mathematical Society, Providence (1994)CrossRefGoogle Scholar
  55. 55.
    Smale, S.: Morse theory and a non-linear generalization of the Dirichlet problem. Ann. Math. 80, 382–396 (1964)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Smale, S.: On the Morse index theorem. J. Math. Mech. 14, 1049–1056 (1965)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Ströhmer, G.: About the morse theory for certain vartional problems. Math. Ann. 270, 275–284 (1985)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Tromba, A.J.: A general approach to Morse theory. J. Differ. Geom. 12, 47–85 (1977)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Uhlenbeck, K.: Morse theory on Banach manifolds. J. Funct. Anal. 10, 430–445 (1972)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Uhlenbeck, K.: The Morse index theorem in Hilbert space. J. Differ. Geom. 8, 555–564 (1973)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Vakhrameev, S.A.: Critical point theory for smooth functions on Hilbert manifolds with singularities and its application to some optimal control problems. J. Sov. Math. 67, 2713–2811 (1993)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Vannella, G.: Morse theory applied to a \(T^2\)-equivriant problem. Topol. Methods Nonlinear Anal. 17, 41–53 (2001)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Viterbo, C.: Indice de Morse des points critiques obtenus par minimax. Ann. Inst. Henri Poincaré 5, 221–225 (1988)MathSciNetCrossRefGoogle Scholar
  64. 64.
    Wang, Z.Q.: Equivariant Morse theory for isolated critical orbits and its applications to nonlinear problems. In: Chern, S.S (ed.) Partial Differential Equations, Proceedings of the Seventh Symposium on Differential Geometry and Differential Equations held in Tianjin, June 23–July 5, 1986. Lecture Notes in Mathematics, vol. 1306, pp. 202–221. Springer, Berlin (1988)Google Scholar
  65. 65.
    Wasserman, G.: Equivariant differential topology. Topology 8, 127–150 (1969)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Wendl, C.: Lectures on Holomorphic Curves in Symplectic and Contact Geometry, math.SG. arXiv:1011.1690
  67. 67.
    Zou, W.M., Schechter, M.: Critical Point Theory and Its Applications. Springer, New York (2006)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Laboratory of Mathematics and Complex Systems, School of Mathematical SciencesBeijing Normal University, Ministry of EducationBeijingThe People’s Republic of China

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