Archive for Rational Mechanics and Analysis

, Volume 134, Issue 3, pp 249–274

Critical points for multiple integrals of the calculus of variations

  • David Arcoya
  • Lucio Boccardo
Article

DOI: 10.1007/BF00379536

Cite this article as:
Arcoya, D. & Boccardo, L. Arch. Rational Mech. Anal. (1996) 134: 249. doi:10.1007/BF00379536

Abstract

In this paper we deal with the existence of critical points of functional defined on the Sobolev space W01,p(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.

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