Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

The Nehari Manifold Approach for a p(x)-Laplacian Problem with Nonlinear Boundary Conditions

  • 62 Accesses

We consider a class of p(x)-Laplacian equations involving nonnegative weight functions with nonlinear boundary conditions. Our technical approach is based on the Nehari manifold, which is similar to the fibering method proposed by Drabek and Pohozaev, together with the recent idea from Brown and Wu.

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

References

  1. 1.

    G. A. Afrouzi and S. H. Rasouli, “A variational approach to a quasilinear elliptic problem involving the p-Laplacian and nonlinear boundary condition,” Nonlin. Anal., 71, 2447–2455 (2009).

  2. 2.

    K. J. Brown and T. F.Wu, “A semilinear elliptic system involving nonlinear boundary condition and sign changing weight function,” J. Math. Anal. Appl., 337, 1326–1336 (2008).

  3. 3.

    P. Drabek and S. I. Pohozaev, “Positive solutions for the p(x)-Laplacian: application of the fibering method,” Proc. Roy. Soc. Edinburgh Sec. A, 127, 721–747 (1997).

  4. 4.

    Y. Chen, S. Levine, and M. Rao, “Variable exponent, linear growth functionals in image restoration,” SIAM J. Appl. Math., 66, No. 4, 1383–1406 (2006).

  5. 5.

    G. Dai, “Infinitely many solutions for a Neumann-type differential inclusion problem involving the p(x)-Laplacian,” Nonlin. Anal., 70, 2297–2305 (2009).

  6. 6.

    G. Dai, “Infinitely many solutions for a hemivariational inequality involving the p(x)-Laplacian,” Nonlin. Anal., 71, 186–195 (2009).

  7. 7.

    G. Dai, “Three solutions for a Neumann-type differential inclusion problem involving the p(x)-Laplacian,” Nonlin. Anal., 70, 3755–3760 (2009).

  8. 8.

    G. Dai, “Infinitely many solutions for a differential inclusion problem in RN involving the p(x)-Laplacian,” Nonlin. Anal., 71, 1116–1123 (2009).

  9. 9.

    X. L. Fan, “On the sub-supersolution methods for p(x)-Laplacian equations,” J. Math. Anal. Appl., 330, 665–682 (2007).

  10. 10.

    X. L. Fan and X. Y. Han, “Existence and multiplicity of solutions for p(x)-Laplacian equations in RN,” Nonlin. Anal., 59, 173–188 (2004).

  11. 11.

    X. L. Fan, J. S. Shen, and D. Zhao, “Sobolev embedding theorems for spaces W k,p(x)(Ω),” J. Math. Anal. Appl., 262, 749–760 (2001).

  12. 12.

    X. L. Fan and Q. H. Zhang, “Existence of solutions for p(x)-Laplacian Dirichlet problems,” Nonlin. Anal., 52, 1843–1852 (2003).

  13. 13.

    X. L. Fan, Q. H. Zhang, and D. Zhao, “Eigenvalues of p(x)-Laplacian Dirichlet problem,” J. Math. Anal. Appl., 302, 306–317 (2005).

  14. 14.

    X. L. Fan and D. Zhao, “On the spaces L p(x) and W k,p(x),” J. Math. Anal. Appl., 263, 424–446 (2001).

  15. 15.

    X. L. Fan, Y. Z. Zhao, and Q. H. Zhang, “A strong maximum principle for p(x)-Laplace equations,” Chinese J. Contemp. Math., 24, No. 3, 277–282 (2003).

  16. 16.

    P. Harjulehto and P. Hasto, “An overview of variable exponent Lebesgue and Sobolev spaces,” in: Future Trends in Geometric Function Theory, Ed. D. Herron, RNC Workshop, Jyvaskyla (2003), pp. 85–93.

  17. 17.

    X. He and W. Zou, “Infinitely many positive solutions for Kirchhoff-type problems,” Nonlin. Anal., 70, 1407–1414 (2009).

  18. 18.

    S. H. Rasouli and G. A. Afrouzi, “The Nehari manifold for a class of concave-convex elliptic systems involving the p-Laplacian and nonlinear boundary condition,” Nonlin. Anal., 73, 3390–3401 (2010).

  19. 19.

    T. F.Wu, “Multiplicity results for a semilinear elliptic equation involving sign-changing weight function,” Rocky Mountain J. Math., 39, No. 3, 995–1011 (2009).

  20. 20.

    T. F. Wu, “A semilinear elliptic problem involving nonlinear boundary condition and sign-changing potential,” Electron. J. Differential Equations, 131, 1–15 (2006).

  21. 21.

    T. F. Wu, “On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function,” J. Math. Anal. Appl., 318, 253–270 (2006).

  22. 22.

    X. Zhang and X. Liu, “The local boundedness and Harnack inequality of p(x)-Laplace equation,” J. Math. Anal. Appl., 332, 209–218 (2007).

Download references

Author information

Correspondence to S. H. Rasouli.

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 1, pp. 92–103, January, 2017.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Rasouli, S.H., Fallah, K. The Nehari Manifold Approach for a p(x)-Laplacian Problem with Nonlinear Boundary Conditions. Ukr Math J 69, 111–125 (2017). https://doi.org/10.1007/s11253-017-1350-6

Download citation