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Transversality Theory with Applications to Differential Equations

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Abstract

The generic existence of Morse functions in a prescribed family of smooth functionals is investigated. The approach is based on arguments involving the transversality theory. The abstract result is applied to semilinear elliptic boundary value problems. One obtains qualitative information concerning the set of solutions.

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References

  1. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)

    MATH  Google Scholar 

  2. Cheng, K.-S., Smoller, J.A.: Symmetry-breaking under small perturbations. Arch. Ration. Mech. Anal. 108, 83–96 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Geba, K.: The Leray-Schauder degree and framed bordism. In: La théorie des points fixes et de ses applications à l’analyse. Presses de l’Université de Montréal, Montréal (1975)

    MATH  Google Scholar 

  4. Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in R n. In: Mathematical Analysis and Applications, Part A, pp. 369–402. Academic, New York (1981)

    Google Scholar 

  5. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)

    MATH  Google Scholar 

  6. Hajlaoui, H., Harrabi, A., Mtiri, F.: Morse indices of solutions for super-linear elliptic PDE’s. Nonlinear Anal. 116, 180–192 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ma, J.: A generalized transversality in global analysis. Pac. J. Math. 236, 357–371 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  9. Mawhin, J.: Problèmes de Dirichlet variationnels non linéaires. Presses de l’Université de Montréal, Montréal (1987)

    MATH  Google Scholar 

  10. Mawhin, J.: Generic properties of nonlinear boundary value problems. In: Differential Equations and Mathematical Physics (Birmingham, AL, 1990), pp. 217–234. Academic, Boston (1992)

    Google Scholar 

  11. Motreanu, D.: Existence for minimization with nonconvex constraints. J. Math. Anal. Appl. 117, 128–137 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  12. Motreanu, D.: Transversality methods in variational and control problems. In: Differential Geometry and Its Applications (Brno, 1989), pp. 265–269. World Scientific, Teaneck (1990)

    Google Scholar 

  13. Motreanu, D.: Generic existence of Morse functions on infinite-dimensional Riemannian manifolds and applications. In: Global Differential Geometry and Global Analysis (Berlin, 1990). Lecture Notes in Mathematics, vol. 1481, pp. 175–184. Springer, Berlin (1991)

    Google Scholar 

  14. Motreanu, D., Motreanu, V.V., Papageorgiou, N.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)

    Book  MATH  Google Scholar 

  15. Palais, R.S.: Seminar on the Atiyah-Singer Index Theorem. Princeton University Press, Princeton (1965)

    Book  MATH  Google Scholar 

  16. Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. Published for the Conference Board of the Mathematical Sciences, Washington, DC. American Mathematical Society, Providence (1986)

    Google Scholar 

  17. Saut, J.-C., Temam, R.: Generic properties of nonlinear boundary value problems. Commun. Partial Differ. Equ. 4, 293–319 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  18. Schwartz, J.T.: Nonlinear Functional Analysis. Gordon and Breach, New York (1969)

    MATH  Google Scholar 

  19. Smale, S.: An infinite dimensional version of Sard’s theorem. Am. J. Math. 87, 861–866 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  20. Struwe, M.: Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems. Manuscr. Math. 32, 335–364 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tanaka, K.: Morse indices at critical points related to the symmetric mountain pass theorem and applications. Commun. Partial Differ. Equ. 14, 99–128 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zeidler, E.: Nonlinear Functional Analysis and Its Applications. IV. Applications to Mathematical Physics. Springer, New York (1988)

    Book  MATH  Google Scholar 

  23. Zeidler, E.: Applied Functional Analysis. Main Principles and Their Applications. Applied Mathematical Sciences, vol. 109. Springer, New York (1995)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Département de Mathématiques, Université de Perpignan, Avenue Paul Alduy 52, 66860, Perpignan, France

    Dumitru Motreanu

  2. Lycée Polyvalent des Métiers du Bâtiment et de l’Énergie “Emmanuel Héré”, 86 Boulevard Maréchal Foch, 54520, Laxou, France

    Viorica Venera Motreanu

Authors
  1. Dumitru Motreanu
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  2. Viorica Venera Motreanu
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Correspondence to Dumitru Motreanu .

Editor information

Editors and Affiliations

  1. Department of Mathematics, National Technical Unversity of Athens, Athens, Greece

    Themistocles M. Rassias

  2. University of Florida, GAINESVILLE, Florida, USA

    Panos M. Pardalos

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Motreanu, D., Motreanu, V.V. (2016). Transversality Theory with Applications to Differential Equations. In: Rassias, T., Pardalos, P. (eds) Essays in Mathematics and its Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-31338-2_7

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