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Critical points for multiple integrals of the calculus of variations. (English) Zbl 0884.58023
This paper is devoted to the existence of critical points for functionals defined on \(W^{1,p}_0 (\Omega)\) by \[ J(u)= \int_\Omega {\mathcal I} (x,u,Du) dx. \] Here \(p>1\), \(\Omega\) is bounded and open in \({\mathbb R}^N\), and \(Du\) denotes the gradient of \(u\). As those functionals may fail to be differentiable, the authors prove a version of the mountain pass lemma which is applicable to such more general situations. Applications are given to the existence and multiplicity of nonnegative critical points. The results are related to earlier ones of Corvellec, Degiovanni and Marzocchi, and of Boccardo, Murat and Puel.

MSC:
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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