Bulletin of the Korean Mathematical Society (대한수학회보)

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ON DISCONTINUOUS ELLIPTIC PROBLEMS INVOLVING THE FRACTIONAL p-LAPLACIAN IN ℝN

  • Kim, In Hyoun (Department of Mathematics Incheon National University) ;
  • Kim, Yun-Ho (Department of Mathematics Education Sangmyung University) ;
  • Park, Kisoeb (Department of Mathematics Incheon National University)
  • Received : 2017.12.31
  • Accepted : 2018.05.23
  • Published : 2018.11.30
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Abstract

We are concerned with the following fractional p-Laplacian inclusion: $$(-{\Delta})^s_pu+V(x){\mid}u{\mid}^{p-2}u{\in}{\lambda}[{\underline{f}}(x,u(x)),\;{\bar{f}}(s,u(x))]$$ in ${\mathbb{R}}^N$, where $(-{\Delta})^s_p$ is the fractional p-Laplacian operator, 0 < s < 1 < p < $+{\infty}$, sp < N, and $f:{\mathbb{R}}^N{\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ is measurable with respect to each variable separately. We show that our problem with the discontinuous nonlinearity f admits at least one or two nontrivial weak solutions. In order to do this, the main tool is the Berkovits-Tienari degree theory for weakly upper semicontinuous set-valued operators. In addition, our main assertions continue to hold when $(-{\Delta})^s_pu$ is replaced by any non-local integro-differential operator.

Keywords

Acknowledgement

Supported by : Incheon National University

References

  1. R. A. Adams and J. J. F. Fournier, Sobolev Spaces, second edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.
  2. A. Ambrosetti and M. Badiale, The dual variational principle and elliptic problems with discontinuous nonlinearities, J. Math. Anal. Appl. 140 (1989), no. 2, 363-373. https://doi.org/10.1016/0022-247X(89)90070-X
  3. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7
  4. M. Badiale and G. Tarantello, Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities, Nonlinear Anal. 29 (1997), no. 6, 639-677. https://doi.org/10.1016/S0362-546X(96)00071-5
  5. G. Barletta, A. Chinni, and D. O'Regan, Existence results for a Neumann problem involving the p(x)-Laplacian with discontinuous nonlinearities, Nonlinear Anal. Real World Appl. 27 (2016), 312-325. https://doi.org/10.1016/j.nonrwa.2015.08.002
  6. B. Barrios, E. Colorado, A. De Pablo, and U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations 252 (2012), no. 11, 6133-6162. https://doi.org/10.1016/j.jde.2012.02.023
  7. T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on ${\mathbb{R}}^N$, Comm. Partial Differential Equations 20 (1995), no. 9-10, 1725-1741. https://doi.org/10.1080/03605309508821149
  8. J. Berkovits and M. Tienari, Topological degree theory for some classes of multis with applications to hyperbolic and elliptic problems involving discontinuous nonlinearities, Dynam. Systems Appl. 5 (1996), no. 1, 1-18.
  9. J. Bertoin, Levy Processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996.
  10. C. Bjorland, L. Caffarelli, and A. Figalli, Non-local gradient dependent operators, Adv. Math. 230 (2012), no. 4-6, 1859-1894. https://doi.org/10.1016/j.aim.2012.03.032
  11. G. Bonanno and G. M. Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl. 2009 (2009), Art. ID 670675, 20 pp.
  12. G. Bonanno and A. Chinni, Discontinuous elliptic problems involving the p(x)-Laplacian, Math. Nachr. 284 (2011), no. 5-6, 639-652. https://doi.org/10.1002/mana.200810232
  13. L. Brasco, E. Parini, and M. Squassina, Stability of variational eigenvalues for the fractional p-Laplacian, Discrete Contin. Dyn. Syst. 36 (2016), no. 4, 1813-1845. https://doi.org/10.3934/dcds.2016.36.1813
  14. F. E. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 1, 1-39. https://doi.org/10.1090/S0273-0979-1983-15153-4
  15. L. Caffarelli, Non-local diffusions, drifts and games, in Nonlinear partial differential equations, 37-52, Abel Symp., 7, Springer, Heidelberg, 2012.
  16. K. C. Chang, The obstacle problem and partial differential equations with discontinuous nonlinearities, Comm. Pure Appl. Math. 33 (1980), no. 2, 117-146. https://doi.org/10.1002/cpa.3160330203
  17. K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), no. 1, 102-129. https://doi.org/10.1016/0022-247X(81)90095-0
  18. X. Chang and Z.-Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations 256 (2014), no. 8, 2965-2992. https://doi.org/10.1016/j.jde.2014.01.027
  19. F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983.
  20. P. Drabek, A. Kufner, and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter & Co., Berlin, 1997.
  21. G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul. 7 (2008), no. 3, 1005-1028. https://doi.org/10.1137/070698592
  22. A. Iannizzotto, S. Liu, K. Perera, and M. Squassina, Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var. 9 (2016), no. 2, 101-125. https://doi.org/10.1515/acv-2014-0024
  23. I.-S. Kim and J.-H. Bae, Elliptic boundary value problems with discontinuous nonlinearities, J. Nonlinear Convex Anal. 17 (2016), no. 1, 27-38.
  24. Y.-H. Kim, Existence of a weak solution for the fractional p-Laplacian equations with discontinuous nonlinearities via the Berkovits-Tienari degree theory, Topol. Methods Nonlinear Anal. 51 (2018), no. 2, 371-388. https://doi.org/10.12775/TMNA.2017.064
  25. N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A 268 (2000), no. 4-6, 298-305. https://doi.org/10.1016/S0375-9601(00)00201-2
  26. J. Lee, J.-M. Kim, and Y.-H. Kim, Existence of weak solutions to a class of Schrodinger type equations involving the fractional p-Laplacian in ${\mathbb{R}}^N$, submitted.
  27. R. Lehrer, L. A. Maia, and M. Squassina, On fractional p-Laplacian problems with weight, Differential Integral Equations 28 (2015), no. 1-2, 15-28.
  28. V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal. 195 (2002), no. 2, 230-238. https://doi.org/10.1006/jfan.2002.3955
  29. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), no. 1, 77 pp.
  30. R. Metzler and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 (2004), no. 31, 161-208. https://doi.org/10.1088/0305-4470/37/1/011
  31. E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521-573. https://doi.org/10.1016/j.bulsci.2011.12.004
  32. K. Perera, M. Squassina, and Y. Yang, Bifurcation and multiplicity results for critical fractional p-Laplacian problems, Math. Nachr. 289 (2016), no. 2-3, 332-342. https://doi.org/10.1002/mana.201400259
  33. R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, in Recent trends in nonlinear partial differential equations. II. Stationary problems, 317-340, Contemp. Math., 595, Amer. Math. Soc., Providence, RI, 2013.
  34. X. Shang, Existence and multiplicity of solutions for a discontinuous problems with critical Sobolev exponents, J. Math. Anal. Appl. 385 (2012), no. 2, 1033-1043. https://doi.org/10.1016/j.jmaa.2011.07.029
  35. M. Struwe, Variational Methods, second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 34, Springer-Verlag, Berlin, 1996.
  36. C. E. Torres Ledesma, Existence and symmetry result for fractional p-Laplacian in ${\mathbb{R}}^n$, Commun. Pure Appl. Anal. 16 (2017), no. 1, 99-113. https://doi.org/10.3934/cpaa.2017004
  37. M. Xiang, B. Zhang, and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian, J. Math. Anal. Appl. 424 (2015), no. 2, 1021-1041. https://doi.org/10.1016/j.jmaa.2014.11.055
  38. Z. Yuan and L. Huang, Non-smooth extension of a three critical points theorem by Ricceri with an application to p(x)-Laplacian differential inclusions, Electron. J. Differential Equations 2015 (2015), no. 232, 16 pp.
  39. E. Zeidler, Nonlinear Functional Analysis and Its Applications. III, translated from the German by Leo F. Boron, Springer-Verlag, New York, 1985.
  40. E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/B, translated from the German by the author and Leo F. Boron, Springer-Verlag, New York, 1990.
  41. B. Zhang and M. Ferrara, Multiplicity of solutions for a class of superlinear non-local fractional equations, Complex Var. Elliptic Equ. 60 (2015), no. 5, 583-595. https://doi.org/10.1080/17476933.2014.959005