AIMS Mathematics, 2020, 5(5): 4466-4481. doi: 10.3934/math.2020287.

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A critical point theorem for a class of non-differentiable functionals with applications

Department of Mathematics, Jining University, Qufu 273155, Shandong, P. R. China

This paper presents a multiplicity theorem for a kind of non-smooth functionals. The proof of this theorem relies on a suitable deformation lemma and the perturbation methods. We also apply this result to prove a multiplicity theorem for elliptic variational-hemivariational inequality problems.
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Keywords critical points; non-smooth functions; deformation; variational-hemivariational inequality; locally Lipschitz continuous

Citation: Yan Ning, Daowei Lu. A critical point theorem for a class of non-differentiable functionals with applications. AIMS Mathematics, 2020, 5(5): 4466-4481. doi: 10.3934/math.2020287

References

  • 1. G. M. Bisci, Some remarks on a recent critical point result of nonsmooth analysis, Le Matematiche, 64 (2009), 97-112.
  • 2. H. Brézis, Analyse Fonctionelle, Théorie et Applications, 1983.
  • 3. H. Brézis, L. Nirenberg, Remarks on finding critical points, Commun. Pur. Appl. Math., 44 (1991), 939-961.    
  • 4. P. Candito, R. Livrea, D. Motreanu, Bounded Palais-Smale sequences for non-differentiable functions, Nonlinear Anal-Theor., 74 (2011), 5446-5454.    
  • 5. K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129.    
  • 6. F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
  • 7. N. Costea, C. Varga, Multiple critical points for non-differentiable parametrized functionals and applications to differential inclusions, J. Global Optim., 56 (2013), 399-416.    
  • 8. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
  • 9. J. Dugundji, Topology, Allyn and Bacon, Boston, 1996.
  • 10. F. Faraci, A. Iannizzotto, Three nonzero periodic solutions for a differential inclusion, Discrete Cont. Dyn. S, 5 (2012), 779-788.
  • 11. A. Iannizzotto, Three critical points for perturbed nonsmooth functionals and applications, Nonlinear Anal-Theor., 72 (2010), 1319-1338.    
  • 12. S. T. Kyritsi, N. S. Papageorgiou, Nonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities, Nonlinear Anal-Theor., 61 (2005), 373-403.    
  • 13. S. T. Kyritsi, N. S. Papageorgiou, An obstacle problem for nonlinear hemivariational inequalities at resonance, J. Math. Anal. Appl., 276 (2002), 292-313.    
  • 14. S. J. Li, M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl., 189 (1995), 6-32.    
  • 15. Z. Li, Y. Shen, Y. Zhang, An application of nonsmooth critical point theory, Topol. Method. Nonl. An., 35 (2010), 203-219.
  • 16. R. Livrea, S. A. Marano, D. Motreanu, Critical points for nondifferential function in presence of splitting, J. Differ. Equations, 226 (2006), 704-725.    
  • 17. A. M. Mao, S. X. Luan, Periodic solutions of an infinite-dimensional Hamiltonian system, Appl. Math. Comput., 201 (2008), 800-804.
  • 18. A. M. Mao, M. Xue, Positive solutions of singular boundary value problems, Acta Math. Sin., 44 (2001), 899-908.
  • 19. A. M. Mao, Y. Chen, Existence and Concentration of Solutions For Sublinear Schrödinger-Poisson Equations, Indian J. Pure Ap. Mat., 49 (2018), 339-348.    
  • 20. S. A. Marano, D. Motreanu, Infinitely many critical points of non-differentiable functions and applications to a Neumann-type problem involving the p-Laplacian, J. Differ. Equations, 182 (2002), 108-120.    
  • 21. S. A. Marano, D. Motreanu, A deformation theorem and some critical point results for nondifferentiale functions, Topol. Method. Nonl. An., 22 (2003), 139-158.
  • 22. D. Motreanu, P. D. Panagiotopoulos, Minimax Theorems and Qualitative properties of the solutions of the Hemivariational Inequalities, 1999.
  • 23. D. Motreanu, V. Radulescu, Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems, Springer Science & Business Media, 2003.
  • 24. P. D. Panagiotopoulos, Hemivariational Inequalities: Applications in Mechanics and Engineering, Springer, Berlin, 1993.
  • 25. A. Szulkin, Minimax principle for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. I. H. Poincare-An., 3 (1986), 77-109.    
  • 26. J. Wang, T. An, F. Zhang, Positive solutions for a class of quasilinear problems with critical growth in $\mathbb{R}^N$, P. Roy. Soc. Edinb. A, 145 (2015), 411-444.
  • 27. Y. Wu, T. An, Existence of periodic solutions for non-autonomous second-order Hamiltonian systems, Electron. J. Differ. Eq., 2013 (2013), 1-13.    

 

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