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Journal of Global Optimization

, Volume 56, Issue 2, pp 399–416 | Cite as

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

  • Nicuşor Costea
  • Csaba Varga
Article

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 : XY, S : XZ 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.

Keywords

Nonsmooth critical point Locally Lipschitz functional p(x)-Laplace operator Multiplicity Differential inclusion Steklov-type boundary condition 

Mathematics Subject Classification (2000)

58K05 47J30 58E05 34A60 47J22 

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Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania
  2. 2.Department of Mathematics and its ApplicationsCentral European UniversityBudapestHungary
  3. 3.Faculty of Mathematics and Computer ScienceBabeş-Bolyai UniversityCluj-NapocaRomania

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