Journal of the Mathematical Society of Japan

Semilinear degenerate elliptic boundary value problems via Morse theory

Kazuaki TAIRA

Full-text: Open access

Abstract

The purpose of this paper is to study a class of semilinear elliptic boundary value problems with degenerate boundary conditions which include as particular cases the Dirichlet and Robin problems. By making use of the Morse and Ljusternik–Schnirelman theories of critical points, we prove existence theorems of non-trivial solutions of our problem. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of semilinear elliptic boundary value problems with degenerate boundary conditions. The results here extend earlier theorems due to Ambrosetti–Lupo and Struwe to the degenerate case.

Article information

Source
J. Math. Soc. Japan, Volume 67, Number 1 (2015), 339-382.

Dates
First available in Project Euclid: 22 January 2015

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1421936556

Digital Object Identifier
doi:10.2969/jmsj/06710339

Mathematical Reviews number (MathSciNet)
MR3304025

Zentralblatt MATH identifier
1315.35104

Subjects
Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 35J20: Variational methods for second-order elliptic equations 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30] 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)

Keywords
semilinear elliptic boundary value problem degenerate boundary condition multiple solution Morse theory Ljusternik–Schnirelman theory

Citation

TAIRA, Kazuaki. Semilinear degenerate elliptic boundary value problems via Morse theory. J. Math. Soc. Japan 67 (2015), no. 1, 339--382. doi:10.2969/jmsj/06710339. https://projecteuclid.org/euclid.jmsj/1421936556


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References

  • R. A. Adams and J. J. F. Fournier, Sobolev Spaces. 2nd ed., Pure Appl. Math. (Amst.), 140, Elsevier; Academic Press, Amsterdam, 2003.
  • H. Amann, A note on degree theory for gradient mappings, Proc. Amer. Math. Soc., 85 (1982), 591–595.
  • H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 7 (1980), 539–603.
  • A. Ambrosetti, On the existence of multiple solutions for a class of nonlinear boundary value problems, Rend. Sem. Mat. Univ. Padova, 49 (1973), 195–204.
  • A. Ambrosetti, Some remarks on the buckling problem for a thin clamped shell, Ricerche Mat., 23 (1974), 161–170.
  • A. Ambrosetti and D. Lupo, On a class of nonlinear Dirichlet problems with multiple solutions, Nonlinear Anal., 8 (1984), 1145–1150.
  • A. Ambrosetti and G. Mancini, Sharp nonuniqueness results for some nonlinear problems, Nonlinear Anal., 3 (1979), 635–645.
  • A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Stud. Adv. Math., 34, Cambridge University Press, Cambridge, 1993.
  • K.-C. Chang, Methods in Nonlinear Analysis, Springer Monogr. Math., Springer-Verlag, Berlin, 2005.
  • D. C. Clark, A variant of the Lusternik–Schnirelman theory, Indiana Univ. Math. J., 22 (1972), 65–74.
  • J. Dugundji, An extension of Tietze's theorem, Pacific J. Math., 1 (1951), 353–367.
  • D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad., 43 (1967), 82–86.
  • J. A. Hempel, Multiple solutions for a class of nonlinear boundary value problems, Indiana Univ. Math. J., 20 (1971), 983–996.
  • L. Hörmander, The Analysis of Linear Partial Differential Operators. III, Pseudo-Differential Operators, 1994 edition, Grundlehren Math. Wiss., 274, Springer-Verlag, Berlin, 1994.
  • L. Ljusternik and L. Schnirelman, Méthodes topologiques dans les problèmes variationneles, Gauthier-Villars, Paris, 1934.
  • A. Marino and G. Prodi, Metodi perturbativi nella teoria di Morse, Boll. Un. Mat. Ital. (4), 11, suppl. (1975), 1–32.
  • M. Morse, The Calculus of Variations in the Large, Amer. Math. Soc. Colloq. Publ., 18, Amer. Math. Soc., Providence, RI, 1934.
  • R. S. Palais, Morse theory on Hilbert manifolds, Topology, 2 (1963), 299–340.
  • R. S. Palais, Lusternik–Schnirelman theory on Banach manifolds, Topology, 5 (1966), 115–132.
  • R. S. Palais and S. Smale, A generalized Morse theory, Bull. Amer. Math. Soc., 70 (1964), 165–172.
  • P. H. Rabinowitz, Variational methods for nonlinear elliptic eigenvalue problems, Indiana Univ. Math. J., 23 (1974), 729–754.
  • J. T. Schwartz, Generalizing the Lusternik–Schnirelman theory of critical points, Comm. Pure Appl. Math., 17 (1964), 307–315.
  • J. T. Schwartz, Nonlinear Functional Analysis, Gordon and Breach, New York, 1969.
  • S. Smale, An infinite dimensional version of Sard's theorem, Amer. J. Math., 87 (1965), 861–866.
  • M. Struwe, A note on a result of Ambrosetti and Mancini, Ann. Mat. Pura Appl., 131 (1982), 107–115.
  • K. Taira, Semigroups, Boundary Value Problems and Markov Processes, Springer Monogr. Math., Springer-Verlag, Berlin, 2004.
  • K. Taira, Degenerate elliptic eigenvalue problems with indefinite weights, Mediterr. J. Math., 5 (2008), 133–162.
  • K. Taira, Degenerate elliptic boundary value problems with asymmetric nonlinearity, J. Math. Soc. Japan, 62 (2010), 431–465.
  • K. Taira, Semilinear degenerate elliptic boundary value problems at resonance, Ann. Univ. Ferrara Sez. VII Sci. Mat., 56 (2010), 369–392.
  • K. Taira, Degenerate elliptic boundary value problems with asymptotically linear nonlinearity, Rend. Circ. Mat. Palermo (2), 60 (2011), 283–308.
  • K. Taira, Semilinear degenerate elliptic boundary value problems via critical point theory, Tsukuba J. Math., 36 (2012), 311–365.
  • K. Thews, A reduction method for some nonlinear Dirichlet problems, Nonlinear Anal., 3 (1979), 795–813.
  • M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall Inc., Englewood Cliffs, NJ, 1967.
  • K. Yosida, Functional Analysis. 6th ed., Grundlehren Math. Wiss., 123, Springer-Verlag, Berlin, Heidelberg, New York, 1980.