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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Chen, Xi / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

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Volume 19, Issue 2

Issues

Multiplicity Results for Degenerate Fractional p-Laplacian Problems with Critical Growth

Li Wang / Jixiu Wang
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  • School of Mathematics and Statistics, Hubei University of Arts and Science, Xiangyang 441053, China
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Published Online: 2018-03-07 | DOI: https://doi.org/10.1515/ijnsns-2017-0195

Abstract

In this paper, we deal with the existence of multiple nontrivial solutions for the following fractional p-Laplacian Kirchhoff problems (R2N|u(x)u(y)|p|xy|N+psdxdy)θ1(Δ)psu=λ|u|θp2u+|u|ps2uinΩ,u=0inRNΩ,

where parameter λ>0 belongs to some left neighbourhood of the eigenvalue of the nonlocal operator

(R2N|u(x)u(y)|p|xy|N+psdxdy)θ1(Δ)ps.

The main feature and difficulty of our problems is the fact that the problem is degenerate.

Keywords: concentration–compactness principle; degenerate fractional Kirchhof-type problem; critical exponent

MSC 2010: 35B33; 35R11; 45C05; 58E05

References

  • [1]

    Corréa F. J. S. A. and Figueiredo G. M., On a p-Kirchhoff equation via Krasnoselskii’S genus, Appl. Math. Lett. 22 (2009), 819–822.CrossrefGoogle Scholar

  • [2]

    Dai G. and Liu D., Infinitely many positive solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal. Appl. 359 (2009), 710–764.Web of ScienceGoogle Scholar

  • [3]

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

  • [4]

    Jin J. and Wu X., Infinitely many radial solutions for Kirchhoff-type problems in ℝN, J. Math. Anal. Appl. 369 (2010), 564–574.CrossrefWeb of ScienceGoogle Scholar

  • [5]

    Kirchhoff G., Mechanik, Teubner, Leipzig, 1883.Google Scholar

  • [6]

    Alves C. O., F. J. S. A. Corréa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), 85–93.CrossrefGoogle Scholar

  • [7]

    Fiscella A. and Valdinoci E., A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal. 94 (2014), 156–170.CrossrefWeb of ScienceGoogle Scholar

  • [8]

    Naimen D., Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent, Nonlinear Differ. Equ. Appl. 21 (2014), 885–914.CrossrefWeb of ScienceGoogle Scholar

  • [9]

    He Y., Li G. and Peng S., Concentrating bound states for Kirchhoff type problem involving critical Sobolev exponents, Adv. Nonlinear Stud. 14 (2014), 483–510.Google Scholar

  • [10]

    Wang J., L. Tian, J. Xu and Zhang F., Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differ. Equ. 253 (2012), 2314–2351.CrossrefWeb of ScienceGoogle Scholar

  • [11]

    Brézis H. and Nirenberg L., Positive solutions of nonlinear elliptic problems involving critical Sobolev exponent, Comm. Pure Appl. Math. 36 (1983), 437–477.CrossrefGoogle Scholar

  • [12]

    Lions P. L., The concentration-compactness principle in the calculus of variations. The limit case, Rev. Mat. Iberoam. 1 (1985), 145–201.Google Scholar

  • [13]

    Perera K., Squassina M. and Yang Y., Birfucation and multiplicity results for critical fractional p-Laplacian problems, Math. Nachr. 289 (2016), 332–342.Google Scholar

  • [14]

    D’Ancona P. and Spagnolo S., Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108} (1992), 247–262.CrossrefGoogle Scholar

  • [15]

    Xiang M., Zhang B. and Ferrara M., Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian, J. Math. Anal. Appl. 424 (2015), 1021– 1041.Google Scholar

  • [16]

    Adams R., Spaces Sobolev, Press Academic, York New, 1975.Google Scholar

  • [17]

    Agarwal R., Perera K. and Zhang Z., On some nonlocal eigenvalue problems, Discrete Contin. Dyn. Syst. S 5 (2012), 707–714.Google Scholar

  • [18]

    Fadell E. and Rabinowitz P., Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), 139–174.CrossrefGoogle Scholar

  • [19]

    Benci V., On critical point theory for indefinite functionals in the presence of symmetrics, Trans. Amer. Math. Soc. 274 (1982), 533–572.CrossrefGoogle Scholar

  • [20]

    Perera K., Squassina M. and Yang Y., Bifurcation and multiplicity results for critical p-Laplacian problems, Topol. Methods Nonlinear Anal.47 (2016), 187–194.Google Scholar

  • [21]

    Xiang M., Zhang B. and Zhang X., A nonhomogeneous fractional p-Kirchhoff type problem involving critical exponent in ℝN, Adv. Stud Nonlinear. 17 (2017), 611–640.Web of ScienceGoogle Scholar

  • [22]

    Perera K., Agarwal R. P. and O’Regan D., Morse theoretic aspects of p-Laplacian type operators, volume 161 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2010.Google Scholar

About the article

Received: 2017-09-04

Accepted: 2018-02-21

Published Online: 2018-03-07

Published in Print: 2018-04-25


The first author was supported by National Natural Science Foundation of China (11561024, 11701178), the second author was was supported by National Natural Science Foundation of China (11501186).


Competing interests The authors declare that they have no competing interests.

Authors contributions All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.


Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 2, Pages 215–222, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2017-0195.

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