Some results on critical groups for a class of functionals defined on Sobolev Banach spaces

Silvia Cingolani; Giuseppina Vannella

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In this paper we deal with the existence of critical points for functionals defined on the Sobolev space $W_{0}^{1, 2} (Ω)$ by $J (v) = \int_{Ω} I (x, v, D v) d x$ , $v \in W_{0}^{1, 2} (Ω)$ , where $Ω$ is a bounded, open subset of $R^{N}$ . Since the differentiability can fail even for very simple examples of functionals defined through multiple integrals of Calculus of Variations, we give a suitable version of the Ambrosetti-Rabinowitz Mountain Pass Theorem, which enables us to the study of critical points for functionals which are not differentiable in all directions....

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