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Nonhomogeneous Hemivariational Inequalities with Indefinite Potential and Robin Boundary Condition

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Abstract

We consider a nonlinear, nonhomogeneous Robin problem with an indefinite potential and a nonsmooth primitive in the reaction term. In fact, the right-hand side of the problem (reaction term) is the Clarke subdifferential of a locally Lipschitz integrand. We assume that asymptotically this term is resonant with respect the principal eigenvalue (from the left). We prove the existence of three nontrivial smooth solutions, two of constant sign and the third nodal. We also show the existence of extremal constant sign solutions. The tools come from nonsmooth critical point theory and from global optimization (direct method).

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References

  1. Panagiotopoulos, P.D.: Hemivariational Inequalities: Applications in Mechanics and Engineering. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  2. Liu, S.: Multiple solutions for coercive \(p\)-Laplacian equations. J. Math. Anal. Appl. 316, 229–236 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Liu, J., Liu, S.: The existence of multiple solutions to quasilinear elliptic equations. Bull. Lond. Math. Soc. 37, 592–600 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Papageorgiou, E., Papageorgiou, N.S.: A multiplicity theorem for problems with the \(p\)-Laplacian. J. Funct. Anal. 244, 63–77 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Aizicovici, S., Papageorgiou, N.S., Staicu, V.: The spectrum and an index formula for the Neumann \(p\)-Laplacian and multiple solutions for problems with crossing nonlinearity. Discret. Contin. Dyn. Syst. 25, 431–456 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Existence of multiple solutions with precise sign information for superlinear Neumann problems. Ann. Mat. Pura Appl. 188, 679–719 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Papageorgiou, N.S., Rădulescu, V.D.: Bifurcation near infinity for the Robin \(p\)-Laplacian. Manuscr. Math. 148, 415–433 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  8. Papageorgiou, N.S., Rădulescu, V.D.: Coercive and noncoercive Neumann problems with an indefinite potential. Forum Math. 28, 545–571 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  9. Marano, S.A., Papageorgiou, N.S.: On a Robin problem with \(p\)-Laplacian and reaction bounded only from above. Monatsch. Math. 180, 317–336 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  10. Marano, S.A., Papageorgiou, N.S.: On a Robin problem with indefinite weight and asymmetric reaction superlinear at \(+\infty \). J. Math. Anal. Appl. 443, 123–145 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  11. Papageorgiou, N.S., Rădulescu, V.D.: Multiplicity theorems for nonlinear nonhomogeneous Robin problems. Rev. Mat. Iberoam. 33, 251–289 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  12. Papageorgiou, N.S., Winkert, P.: Nonlinear Robin problems with a reaction of arbitrary growth. Ann. Mat. Pura Appl. 195, 1207–1235 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gasinski, L., Papageorgiou, N.S.: Nonlinear Analysis. Chapman & Hall/CRC, Boca Raton (2006)

    MATH  Google Scholar 

  14. Clarke, F.C.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  15. Chang, K.C.: Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80, 102–129 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gasinski, L., Papageorgiou, N.S.: Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems. Chapman & Hall/CRC, Boca Raton (2005)

    MATH  Google Scholar 

  17. Rădulescu, V.D.: Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations. Hindawi Publishing Corporation, New York (2008)

    Book  MATH  Google Scholar 

  18. Lieberman, G.: The natural generalization of the natural conditions of Ladyszhenskaya and Uraltseva for elliptic equations. Commun. Partial Differ. Equ. 16, 311–361 (1991)

    Article  MATH  Google Scholar 

  19. Pucci, P., Serrin, J.: The Maximum Principle. Birkhäuser, Basel (2007)

    MATH  Google Scholar 

  20. Cherfils, L., Ilyasov, Y.: On the stationary solutions for generalized reaction-diffusion equations with \(p\) & \(q\) Laplacian. Commun. Pure Appl. Anal. 4, 9–22 (2005)

    MATH  MathSciNet  Google Scholar 

  21. Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Nodal solutions for \((p,2)\)-equations. Trans. Am. Math. Soc. 367, 7342–7372 (2015)

    MATH  MathSciNet  Google Scholar 

  22. Cingolani, S., Degiovanni, M.: Nontrivial solutions for \(p\)-Laplacian equations with right-hand side having \(p\)-linear growth. Commun. Partial Differ. Equ. 30, 1191–1203 (2005)

    Article  MATH  Google Scholar 

  23. Papageorgiou, N.S., Rădulescu, V.D.: Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance. Appl. Math. Optim. 69, 393–430 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  24. Papageorgiou, N.S., Rădulescu, V.D., Repovš, D.D.: On a class of parametric \((p,2)\)-equations. Appl. Math. Optim. 75, 193–228 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  25. Sun, M.: Multiplicity of solutions for a class of quasilinear elliptic equations at resonance. J. Math. Anal. Appl. 386, 661–668 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  26. Sun, M., Zhang, M., Su, J.: Critical groups at zero multiple solutions for a quasilinear elliptic equation. J. Math. Anal. Appl. 428, 696–712 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  27. Gasinski, L., Papageorgiou, N.S.: Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential. Set Valued Variat. Anal. 20, 417–443 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  28. Papageorgiou, N.S., Rădulescu, V.D.: Nonlinear nonhomogeneous Robin problems with superlinear reaction term. Adv. Nonlinear Stud. 16, 737–764 (2016)

    MATH  MathSciNet  Google Scholar 

  29. Gasinski, L., Papageorgiou, N.S.: Exercises in Analysis Part 2: Nonlinear Analysis. Springer, Heidelberg (2016)

    Book  MATH  Google Scholar 

  30. Mugnai, D., Papageorgiou, N.S.: Resonant nonlinear Neumann problems with indefinite weight. Ann. Sc. Norm. Super. Pisa Cl. Sci. Ser. V 11(4), 729–788 (2012)

    MATH  MathSciNet  Google Scholar 

  31. Papageorgiou, N.S., Rădulescu, V.D.: Multiple solutions with precise sign for nonlinear parametric Robin problems. J. Differ. Equ. 256, 393–430 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  32. Diaz, J.I., Saa, J.E.: Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires. C.R. Acad. Sci. Paris Sér. I 305, 521–524 (1987)

    MATH  MathSciNet  Google Scholar 

  33. Filippakis, M., Papageorgiou, N.S.: Multiple constant sign and nodal solutions for nonlinear elliptic equations. J. Differ. Equ. 245, 843–870 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  34. Papageorgiou, N.S., Rădulescu, V.D., Repovš, D.D.: Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential. Discret. Contin. Dyn. Syst. 37, 2589–2618 (2017)

    Article  MATH  MathSciNet  Google Scholar 

  35. Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis Volume I: Theory. Kluwer Academic Publishers, Dordrecht (1997)

    Book  MATH  Google Scholar 

  36. Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints. Mem. Am. Math. Soc. 196(915), vi+70 (2008)

  37. Papageorgiou, N.S., Rădulescu, V.D.: Solutions with sign information for nonlinear nonhomogeneous elliptic equations. Topol. Methods Nonlinear Anal. 45, 575–600 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  38. Corvellec, J.-N.: On the second deformation lemma. Topol. Methods Nonlinear Anal. 17, 55–66 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

The authors wish to thank the two anonymous reviewers and the Editor-in-Chief for their corrections and remarks. This research was supported by the Slovenian Research Agency Grants P1-0292, J1-8131, and J1-7025. V. D. Rădulescu acknowledges the support through a grant of the Ministry of Research and Innovation, CNCS–UEFISCDI, Project number PN-III-P4-ID-PCE-2016-0130, within PNCDI III.

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Authors and Affiliations

  1. Department of Mathematics, National Technical University, Zografou Campus, 15780, Athens, Greece

    Nikolaos S. Papageorgiou

  2. Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Vicenţiu D. Rădulescu

  3. Department of Mathematics, University of Craiova, Street A.I. Cuza No 13, 200585, Craiova, Romania

    Vicenţiu D. Rădulescu

  4. Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, 1000, Ljubljana, Slovenia

    Dušan D. Repovš

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  1. Nikolaos S. Papageorgiou
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  2. Vicenţiu D. Rădulescu
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Correspondence to Vicenţiu D. Rădulescu.

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Papageorgiou, N.S., Rădulescu, V.D. & Repovš, D.D. Nonhomogeneous Hemivariational Inequalities with Indefinite Potential and Robin Boundary Condition. J Optim Theory Appl 175, 293–323 (2017). https://doi.org/10.1007/s10957-017-1173-5

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  • DOI: https://doi.org/10.1007/s10957-017-1173-5

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