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
Article

Abstract

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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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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