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Critical points for non differentiable functionals. (English) Zbl 0892.58013

The paper deals with the existence of critical points for integral functionals defined in the Sobolev space \(W_0^{1,p}\) by \[ J(u)= \int_\Omega j(x,u,Du) dx-\int_\Omega G(x,u)dx \] where \(\Omega\subset \mathbb{R}^N\) is a bounded domain and \(j\) satisfies some “ellipticity” conditions.
It is known that, in general, \(J\) is not Fréchet differentiable on \(W_0^{1,p}\) but only Gâteux-derivable along directions from \(W_0^{1,p} \cap L_\infty\).
Under several restrictions on \(G\), the author proves the existence of infinitely many (pairs of) critical points of \(J\). The basic tool is a variant of the ‘Critical point theory for continuous functionals’ developed by M. Degiovanni [see J. Corvellec, M. Degiovanni and M. Marzocchi, Topol. Methods Nonlinear Anal. 1, No. 1, 151-171 (1993; Zbl 0789.58021)].
Reviewer: V.Moroz (Minsk)

MSC:

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

Citations:

Zbl 0789.58021
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