Skip to main content
Log in

Multiple Solutions of Neumann Problems: An Orlicz–Sobolev Space Setting

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

In the present paper, we establish the range of two parameters for which a non-homogeneous boundary value problem admits at least three weak solutions. The proof of the main results relies on recent variational principles due to Ricceri.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Chen, Y., Levine, S., Rao, R.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66, 1383–1406 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chabrowski, J., Fu, Y.: Existence of solutions for \(p(x)\)-Laplacian problems on a bounded domain. J. Math. Anal. Appl. 306, 604–618 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fan, X.: Solutions for \(p(x)\)-Laplacian Dirichlet problems with singular coefficients. J. Math. Anal. Appl. 312, 464–477 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fan, X., Zhang, Q., Zhao, D.: Eigenvalues of \(p(x)\)-Laplacian Dirichlet problem. J. Math. Anal. Appl. 302, 306–317 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Rădulescu, V.: Nonlinear elliptic equations with variable exponent: old and new. Nonlinear Anal. 121, 336–369 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Rădulescu, V., Repovš, D.: Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Monographs and Research Notes in Mathematics. CRC Press, Taylor & Francis Group, Boca Raton (2015)

    Book  Google Scholar 

  7. Ružička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Berlin (2000)

    MATH  Google Scholar 

  8. Halsey, T.C.: Electrorheological fluids. Science 258, 761–766 (1992)

    Article  Google Scholar 

  9. Pfeiffer, C., Mavroidis, C., Bar-Cohen, Y., Dolgin, B.: Electrorheological Fluid Based Force Feedback Device. In: Proceedings 1999 SPIE telemanipulator and telepresence technologies VI Conference. Boston, vol. 3840, pp. 88–99 (1999)

  10. Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1991)

    MATH  Google Scholar 

  11. Afrouzi, G.A., Hadjian, A., Heidarkhani, S.: Steklov problems involving the \(p(x)\)-Laplacian. Electr. J. Diff. Equ. 134, 1–11 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Allaoui, M., Amrouss, A.R., Ourraoui, A.: Existence and multiplicity of solutions for a Steklov problem involving \(p(x)\)-Laplace operator. Electr. J. Diff. Equ. 132, 1–12 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Demarque, R., Miyagaki, O.: Radial solutions of inhomogeneous fourth order elliptic equations and weighted Sobolev embeddings. Adv. Nonlinear Anal. 4, 135–151 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Deng, S.G.: Eigenvalues of the \(p(x)\)-Laplacian Steklov problem. J. Math. Anal. Appl. 339, 925–937 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tiwari, S.: \(N\)-Laplacian critical problem with discontinuous nonlinearities. Adv. Nonlinear Anal. 4, 109–121 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Torne, O.: Steklov problem with an indefinite weight for the \(p\)-Laplacian. Electron. J. Differ. Equ. 87, 1–8 (2005)

    MathSciNet  MATH  Google Scholar 

  17. Pucci, P., Serrin, J.: Extensions of the mountain pass theorem. J. Funct. Anal. 59, 185–210 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pucci, P., Serrin, J.: A mountain pass theorem. J. Differ. Equ. 60, 142–149 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ricceri, B.: On a three critical points theorem. Arch. Math. (Basel) 75, 220–226 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ricceri, B.: A further refinement of a three critical points theorem. Nonlinear Anal. 74, 7446–7454 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kristály, A., Rădulescu, V., Varga, C.: Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, No. 136. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  22. Beirao da Veiga, H.: On nonlinear potential theory, and regular boundary points, for the \(p\)-Laplacian in \(N\) space variables. Adv. Nonlinear Anal. 3, 45–67 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Heidarkhani, S., Afrouzi, G.A., Hadjian, A.: Multiplicity results for elliptic problems with variable exponent and non-homogeneous Neumann conditions. Math. Methods Appl. Sci. (to appear)

  24. Molica Bisci, G., Rădulescu, V.: Multiple symmetric solutions for a Neumann problem with lack of compactness. CR. Acad. Sci. Paris, Ser. I 351, 37–42 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Molica Bisci, G., Repovš, D.: Multiple solutions for elliptic equations involving a general operator in divergence form. Ann. Acad. Sci. Fenn. Math. 39, 259–273 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ouaro, S., Ouedraogo, A., Soma, S.: Multivalued problem with Robin boundary condition involving diffuse measure data and variable exponent. Adv. Nonlinear Anal. 3, 209–235 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Bonanno, G., Candito, P.: Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities. J. Differ. Equ. 244, 3031–3059 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  29. Diening, L.: Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129, 657–700 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notesin Mathematics, vol. 1034. Springer, Berlin (1983)

  31. Mihăilescu, M., Rădulescu, V.: Neumann problems associated to nonhomogeneous differential operators in Orlicz-Sobolev space. Ann. Inst. Fourier Grenoble 6, 2087–2111 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kristály, A., Mihăilescu, M., Rădulescu, V.: Two non-trivial solutions for a non-homogeneous Neumann problem: an Orlicz-Sobolev space setting. Proc. R. Soc. Edinb. Sect. A 139, 367–379 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Fan, X., Zhao, D.: On the spaces \(L^{p(x)}(\Omega )\) and \(W^{m, p(x)}(\Omega )\). J. Math. Anal. Appl. 263, 424–446 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}(\Omega )\) and \(W^{m, p(x)}(\Omega )\). Czechoslov. Math. J. 41, 592–618 (1991)

    MATH  Google Scholar 

Download references

Acknowledgments

V. Rădulescu acknowledges the support through Grant Advanced Collaborative Research Projects CNCS-PCCA-23/2014.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Vicenţiu D. Rădulescu.

Additional information

Communicated by Norhashidah Hj. Mohd. Ali.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Afrouzi, G.A., Rădulescu, V.D. & Shokooh, S. Multiple Solutions of Neumann Problems: An Orlicz–Sobolev Space Setting. Bull. Malays. Math. Sci. Soc. 40, 1591–1611 (2017). https://doi.org/10.1007/s40840-015-0153-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-015-0153-x

Keywords

Mathematics Subject Classification

Navigation