Skip to main content
Log in

Critical points for multiple integrals of the calculus of variations

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript


In this paper we deal with the existence of critical points of functional defined on the Sobolev space W 1,p0 (Ω), p>1, by

$$J(u) = \int\limits_\Omega {\vartheta (x,u,Du)dx,} {\text{ }}$$

where Ω is a bounded, open subset of ℝN. Even for very simple examples in ℝN 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Ambrosetti, A. Critical points and nonlinear variational problems. Supplement au Bulletin Soc. Math. France, Mémoire n. 49, 1992.

  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. Ambrosetti, A. & Rabinowitz, P. H., Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349–381.

    Google Scholar 

  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. 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.

  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.

  7. Arcoya, D. & Calahorrano, M., Multivalued non-positone problems. Rend. Mat. Acc. Lincei, Ser. 9, 1 (1990), 117–123.

    Google Scholar 

  8. Arcoya, D., Drábek, P. & Zertiti, A., Minimization problem for some degenerated functional: nonnegative and bounded solutions. Preprint (1994).

  9. Aubin, J. P. & Ekeland, I., Applied nonlinear analysis. Wiley, Interscience, New York, 1984.

    Google Scholar 

  10. Benci, V. & Rabinowitz, P. H., Critical point theorems for indefinite functional. Invent. Math. 52 (1979), 241–273.

    Google Scholar 

  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. 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. 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. Brezis, H. & Nirenberg, L. Remarks on finding critical points. Comm. Pure Appl. Math. 44 (1991), 939–963.

    Google Scholar 

  15. Canino, A., Multiplicity of solutions for quasilinear elliptic equations. Preprint (1994).

  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. Dacorogna, B., Direct methods in the calculus of variations. Springer-Verlag, 1989.

  18. De Figueiredo, D. G., The Ekeland variational principle with applications and detours. Springer-Verlag, 1989.

  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. Ekeland, I., Nonconvex minimization problems. Bull. Amer. Math. Soc. (NS) 1 (1979), 443–474.

    Google Scholar 

  21. Ladyženskaya, O. A. & Uralceva, N. N., Linear and quasilinear elliptic equations. Academic Press, New York, 1968.

    Google Scholar 

  22. Leoni, G., Existence of solutions for holonomic dynamical systems with homogeneous boundary conditions. Nonlinear Anal. 23 (1994), 427–445.

    Google Scholar 

  23. Ma, Li, On nonlinear eigen-problems of quasilinear elliptic operators. J. Partial Diff. Eqs. 4 (1991), 56–72.

    Google Scholar 

  24. Mawhin, J. & Willem, M., Critical point theory and Hamiltonian systems. Springer-Verlag, 1989.

  25. Morrey, C. B., Multiple integrals in the calculus of variations. Springer-Verlag, 1966.

  26. Pucci, P. & Serrin, J., A mountain pass theorem. J. Diff. Eqs. 60 (1985), 142–149.

    Google Scholar 

  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. Struwe, M., Variational methods. Springer-Verlag, 1990.

Download references

Author information

Authors and Affiliations


Additional information

Communicated by J. Serrin

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arcoya, D., Boccardo, L. Critical points for multiple integrals of the calculus of variations. Arch. Rational Mech. Anal. 134, 249–274 (1996).

Download citation

  • Accepted:

  • Issue Date:

  • DOI: