Archive for Rational Mechanics and Analysis

, Volume 134, Issue 3, pp 249–274 | Cite as

Critical points for multiple integrals of the calculus of variations

  • David Arcoya
  • Lucio Boccardo


In this paper we deal with the existence of critical points of functional defined on the Sobolev space W 0 1,p (Ω), p>1, by
$$J(u) = \int\limits_\Omega {\vartheta (x,u,Du)dx,} {\text{ }}$$
where Ω is a bounded, open subset of ℝ N . Even for very simple examples in ℝ N the differentiability of J(u) can fail. To overcome this difficulty we prove a suitable version of the Ambrosetti-Rabinowitz Mountain Pass Theorem applicable to functionals which are not differentiable in all directions. Existence and multiplicity of nonnegative critical points are also studied through the use of this theorem.


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  1. 1.
    Ambrosetti, A. Critical points and nonlinear variational problems. Supplement au Bulletin Soc. Math. France, Mémoire n. 49, 1992.Google Scholar
  2. 2.
    Ambrosetti, A. & Prodi, G., On the inversion of some differentiate mappings with singularities between Banach spaces. Ann. Math. Pura Appl. 93 (1972), 231–246.Google Scholar
  3. 3.
    Ambrosetti, A. & Rabinowitz, P. H., Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349–381.Google Scholar
  4. 4.
    Anane, A., Simplicité et isolation de la première valeur propre du-laplacien avec poids. C. R. Acad. Sci. Paris 1 (1988), 341–348.Google Scholar
  5. 5.
    Arcoya, D. & Boccardo, L., Nontrivial solutions to some nonlinear equations via minimization. To appear in Proc. International Conference on Nonlinear P.D.E., Erice (Italy), May 1992.Google Scholar
  6. 6.
    Arcoya, D. & Boccardo, L., A min-max theorem for multiple integrals of the Calculus of Variations and applications. To appear in Rend. Mat. Acc. Lincei.Google Scholar
  7. 7.
    Arcoya, D. & Calahorrano, M., Multivalued non-positone problems. Rend. Mat. Acc. Lincei, Ser. 9, 1 (1990), 117–123.Google Scholar
  8. 8.
    Arcoya, D., Drábek, P. & Zertiti, A., Minimization problem for some degenerated functional: nonnegative and bounded solutions. Preprint (1994).Google Scholar
  9. 9.
    Aubin, J. P. & Ekeland, I., Applied nonlinear analysis. Wiley, Interscience, New York, 1984.Google Scholar
  10. 10.
    Benci, V. & Rabinowitz, P. H., Critical point theorems for indefinite functional. Invent. Math. 52 (1979), 241–273.Google Scholar
  11. 11.
    Boccardo, L., Gallouet, T. & Murat, F., A unified presentation of two existence results for problems with natural growth. Pitman Research Notes in Mathematics 296 (1993), 127–137.Google Scholar
  12. 12.
    Boccardo, L., Murat, F. & Puel, J. P., Résultats d'existence pour certains problèmes elliptiques quasilinéaires. Ann. Scuola Norm. Sup. Pisa. 11 (1984), 213–235.Google Scholar
  13. 13.
    Boccardo, L., Murat, F. & Puel, J. P., Existence of bounded solutions for nonlinear unilateral problems. Ann. Mat. Pura Appl. 152 (1988), 183–196.Google Scholar
  14. 14.
    Brezis, H. & Nirenberg, L. Remarks on finding critical points. Comm. Pure Appl. Math. 44 (1991), 939–963.Google Scholar
  15. 15.
    Canino, A., Multiplicity of solutions for quasilinear elliptic equations. Preprint (1994).Google Scholar
  16. 16.
    Chang, K. C., Variational methods for nondifferentiable functional and their applications to partial differential equations. J. Math. Anal. Appl. 80 (1981), 102–129.Google Scholar
  17. 17.
    Dacorogna, B., Direct methods in the calculus of variations. Springer-Verlag, 1989.Google Scholar
  18. 18.
    De Figueiredo, D. G., The Ekeland variational principle with applications and detours. Springer-Verlag, 1989.Google Scholar
  19. 19.
    De Figueiredo, D. G. & Solimini, S., A variational approach to superlinear elliptic problems. Comm. Partial Diff. Eqs. 9 (1984), 699–717.Google Scholar
  20. 20.
    Ekeland, I., Nonconvex minimization problems. Bull. Amer. Math. Soc. (NS) 1 (1979), 443–474.Google Scholar
  21. 21.
    Ladyženskaya, O. A. & Uralceva, N. N., Linear and quasilinear elliptic equations. Academic Press, New York, 1968.Google Scholar
  22. 22.
    Leoni, G., Existence of solutions for holonomic dynamical systems with homogeneous boundary conditions. Nonlinear Anal. 23 (1994), 427–445.Google Scholar
  23. 23.
    Ma, Li, On nonlinear eigen-problems of quasilinear elliptic operators. J. Partial Diff. Eqs. 4 (1991), 56–72.Google Scholar
  24. 24.
    Mawhin, J. & Willem, M., Critical point theory and Hamiltonian systems. Springer-Verlag, 1989.Google Scholar
  25. 25.
    Morrey, C. B., Multiple integrals in the calculus of variations. Springer-Verlag, 1966.Google Scholar
  26. 26.
    Pucci, P. & Serrin, J., A mountain pass theorem. J. Diff. Eqs. 60 (1985), 142–149.Google Scholar
  27. 27.
    Rabinowitz, P. H., Minimax methods in critical point theory with applications to differential equations. CBMS Regional Conference Series Math. 65, Amer. Math. Soc., Providence, 1986.Google Scholar
  28. 28.
    Struwe, M., Variational methods. Springer-Verlag, 1990.Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • David Arcoya
    • 1
    • 2
  • Lucio Boccardo
    • 1
    • 2
  1. 1.Departamento de Análisis MatemáticoUniversidad de GranadaGranada
  2. 2.Dipartimento di MatematicaUniversità di Roma IRoma

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