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Critical points for multiple integrals of the calculus of variations

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Abstract

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.

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Communicated by J. Serrin

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Arcoya, D., Boccardo, L. Critical points for multiple integrals of the calculus of variations. Arch. Rational Mech. Anal. 134, 249–274 (1996). https://doi.org/10.1007/BF00379536

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