Skip to main content
SpringerLink
Log in

Existence of weak solutions for a nonlinear problem involving \(p(\cdot )\)-Laplacian operator with mixed boundary conditions

  • Original Research Paper
  • Published:
The Journal of Analysis Aims and scope Submit manuscript

Abstract

In this paper, we consider a mixed boundary value problem to a class of quasi-linear elliptic operators containing \(p(\cdot )\)-Laplacian operator. More precisely we consider the problem with the Dirichlet condition on a part of the boundary and the Steklov boundary condition on an another part of the boundary. We show the existence of at least one non-trivial weak solution under some hypotheses and the existence of infinitely many weak solutions under some hypotheses.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Allaoui, M., A.R. El Amorousse, and A. Ourraoui. 2012. Existence and multiplicity of solutions for a Steklov problem involving the \(p(x)\)-Laplace operator. Electronic Journal of Differential Equations 132: 1–12.

    MathSciNet  Google Scholar 

  2. Aramaki, J. 2018. Existence of weak solution for a class of abstract coupling system associated with stationary electromagnetic system. Taiwanese Journal of Mathematics 22 (3): 741–765.

    Article  MathSciNet  Google Scholar 

  3. Aramaki, J. 2019. Existence of weak solutions to stationary and evolutionary Maxwell–Stokes type problems and the asymptotic behavior of the solution. Advances in Mathematical Sciences and Applications 28 (1): 29–57.

    MathSciNet  MATH  Google Scholar 

  4. Aramaki, J. 2019. Existence and regularity of a weak solution to a class of systems in a multi-connected domain. Journal of Partial Differential Equations 32 (1): 1–19.

    Article  MathSciNet  Google Scholar 

  5. Avci, M. 2013. Existence and multiplicity of solutions for Dirichlet problems involving the \(p(x)\)-Laplace operator. Electronic Journal of Differential Equations 14: 1–9.

    MathSciNet  MATH  Google Scholar 

  6. Ayoujil, A. 2017. Existence results for Steklov problem involving the \(p(x)\)-Laplacian. Complex Variables and Elliptic Equations 63: 1675–1686.

    Google Scholar 

  7. Ciarlet, P.G. 2013. Linear and nonlinear functional analysis with applications. Philadelphia: SIAM.

    MATH  Google Scholar 

  8. Ciarlet, P.G., and G. Dinca. 2011. A Poincaré inequality in a Sobolev space with a variable exponent. Chinese Annals of Mathematics Series B 32B (3): 333–342.

    Article  Google Scholar 

  9. Deng, S.G. 2008. Existence of the \(p(x)\)-Laplacian Steklov problem. Journal of Mathematical Analysis and Applications 339: 925–937.

    Article  MathSciNet  Google Scholar 

  10. Diening, L. 2002. Theoretical and numerical results for electrorheological fluids, Ph.D. thesis, University of Frieburg, Germany.

  11. Diening, L., P. Harjulehto, P. Hästö, and M. Růžička. 2011. Lebesgue and Sobolev Spaces with Variable Exponent, Lecture Notes in Mathematics, vol. 2017. Berlin: Springer.

    Book  Google Scholar 

  12. Fan, F. 2005. Solutions for \(p(x)\)-Dirichlet problems with singular coefficients. Journal of Mathematical Analysis and Applications 312: 749–760.

    Article  MathSciNet  Google Scholar 

  13. Fan, X.L., and Q.H. Zhang. 2003. Existence of solutions for \(p(x)\)-Laplacian Dirichlet problem. Nonlinear Analysis 52: 1843–1852.

    Article  MathSciNet  Google Scholar 

  14. Fan, X.L., and D. Zhao. 2001. On the spaces \(L^{p(x)}(\Omega )\) and \(W^{m, p(x)}(\Omega )\). Journal of Mathematical Analysis and Applications 263: 424–446.

    Article  MathSciNet  Google Scholar 

  15. Fan, X.L., Q. Zhang, and D. Zhao. 2015. Eigenvalues of \(p(x)\)-Laplacian Dirichlet problem. Journal of Mathematical Analysis and Applications 302: 306–317.

    Article  MathSciNet  Google Scholar 

  16. Halsey, T.C. 1992. Electrorheological fluids. Science 258: 761–766.

    Article  Google Scholar 

  17. Kováčik, O., and J. Rákosník. 1991. On spaces \(L^{p(x)}(\Omega )\) and \(W^{k, p(x)}(\Omega )\). Czechoslovak Mathematical Journal 41 (116): 592–618.

    Article  MathSciNet  Google Scholar 

  18. Mihăilescu, M., and V. Rădulescu. 2006. A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids. Proceeding of the Royal Society A 462: 2625–2641.

    MathSciNet  MATH  Google Scholar 

  19. Růžička, M. 2000. Electrotheological fluids: modeling and mathematical theory, lecture notes in mathematics, vol. 1784. Berlin: Springer.

    Google Scholar 

  20. Sobolev, S. 1963. Applications of functional analysis in mathematical physics, translation of mathematical monographs, vol. 7. Providence, R. I.: AMS.

    Book  Google Scholar 

  21. Wei, Z. and Z. Chen 2012. Existence results for the \(p(x)\)-Laplacian with nonlinear boundary condition. ISRN Applied Mathematics 727398

  22. Willem, M. 1996. Minimax theorems. Boston, Basel, Berlin: Birkhäuser.

    Book  Google Scholar 

  23. Yao, J. 2008. Solutions for Neumann boundary value problem involving \(p(x)\)-Laplace operators. Nonlinear Analysis 68 (5): 1271–1283.

    Article  MathSciNet  Google Scholar 

  24. Yücedag, Z. 2015. Solutions of nonlinear problems involving \(p(x)\)-Laplacian operator. Advances in Nonlinear Analysis 1–9.

  25. Yücedag, Z. 2019. Existence results for Steklov problem with nonlinear boundary condition. Middle East Journal of Science 5 (2): 146–154.

    Article  Google Scholar 

  26. Zeidler, E. 1990. Nonlinear functional analysis and its applications vol. I: fixed-point theorems and vol. II/B: nonlinear monotone operators. Now York: Springer.

    Google Scholar 

  27. Zhao, J.F. 1991. Structure Theory in Banach Space. Wuhan (in Chinese): Wuhan Univ. Press.

    Google Scholar 

  28. Zhao, D., W.J. Qiang, and X.L. Fan. 1997. On generalized Orlicz space \(L^{p(x)}(\Omega )\). Journal of Gansu Science 9 (2): 1–7 ((in Chinese)).

    Google Scholar 

  29. Zhikov, V.V. 1987. Averaging of functionals of the calculus of variation and elasticity theory. Mathematics of the USSR Izvestiya 29: 33–66.

    Article  Google Scholar 

Download references

Acknowledgements

The author would like to thank the anonymous referee(s) for useful comments and suggestions.

Author information

Authors and Affiliations

  1. Division of Science, Tokyo Denki University, Hatoyama-machi, Saitama, 350-0394, Japan

    Junichi Aramaki

Authors
  1. Junichi Aramaki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Junichi Aramaki.

Ethics declarations

Conflict of interest

The author declares that he has no conflict of interest.

Additional information

Communicated by Samy Ponnusamy.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aramaki, J. Existence of weak solutions for a nonlinear problem involving \(p(\cdot )\)-Laplacian operator with mixed boundary conditions. J Anal 30, 1283–1304 (2022). https://doi.org/10.1007/s41478-022-00408-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41478-022-00408-y

Keywords

Mathematics Subject Classification

Search

Navigation