Multiple critical points for non-differentiable parametrized functionals and applications to differential inclusions

Nicuşor Costea () and Csaba Varga ()

Journal of Global Optimization, 2013, vol. 56, issue 2, 399-416

Abstract: In this paper we deal with a class of non-differentiable functionals defined on a real reflexive Banach space X and depending on a real parameter of the form $${\mathcal{E}_\lambda(u)=L(u)-(J_1\circ T)(u)-\lambda (J_2\circ S)(u)}$$ , where $${L:X \rightarrow \mathbb R}$$ is a sequentially weakly lower semicontinuous C 1 functional, $${J_1:Y\rightarrow\mathbb R, J_2:Z\rightarrow \mathbb R}$$ (Y, Z Banach spaces) are two locally Lipschitz functionals, T : X → Y, S : X → Z are linear and compact operators and λ > 0 is a real parameter. We prove that this kind of functionals posses at least three nonsmooth critical points for each λ > 0 and there exists λ* > 0 such that the functional $${\mathcal{E}_{\lambda^\ast}}$$ possesses at least four nonsmooth critical points. As an application, we study a nonhomogeneous differential inclusion involving the p(x)-Laplace operator whose weak solutions are exactly the nonsmooth critical points of some “energy functional” which satisfies the conditions required in our main result. Copyright Springer Science+Business Media, LLC. 2013

Keywords: Nonsmooth critical point; Locally Lipschitz functional; p(x)-Laplace operator; Multiplicity; Differential inclusion; Steklov-type boundary condition; 58K05; 47J30; 58E05; 34A60; 47J22 (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10898-011-9801-3

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