• Kim, Yun-Ho (Department of Mathematics Education Sangmyung University)
  • Received : 2019.10.15
  • Accepted : 2020.02.13
  • Published : 2020.11.01


We are concerned with the following elliptic equations: $$\{(-{\Delta})^s_pu={\lambda}f(x,u)\;{\text{in {\Omega}}},\\u=0\;{\text{on {\mathbb{R}}^N{\backslash}{\Omega}},$$ where λ are real parameters, (-∆)sp is the fractional p-Laplacian operator, 0 < s < 1 < p < + ∞, sp < N, and f : Ω × ℝ → ℝ satisfies a Carathéodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters λ for which our problem admits at least one or two nontrivial weak solutions when the nonlinearity f has the subcritical growth condition. In addition, under adequate conditions, we establish an apriori estimate in L(Ω) of any possible weak solution by applying the bootstrap argument.


  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 P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349-381.
  3. J.-H. Bae and Y.-H. Kim, Critical points theorems via the generalized Ekeland variational principle and its application to equations of p(x)-Laplace type in $R^N$, Taiwanese J. Math. 23 (2019), no. 1, 193-229.
  4. 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.
  5. 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.
  6. J. Bertoin, Levy Processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996.
  7. Z. Binlin, G. Molica Bisci, and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity 28 (2015), no. 7, 2247-2264.
  8. C. Bjorland, L. Caffarelli, and A. Figalli, Non-local gradient dependent operators, Adv. Math. 230 (2012), no. 4-6, 1859-1894.
  9. G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal. 75 (2012), no. 5, 2992-3007.
  10. G. Bonanno, Relations between the mountain pass theorem and local minima, Adv. Nonlinear Anal. 1 (2012), no. 3, 205-220.
  11. G. Bonanno and A. Chinni, Discontinuous elliptic problems involving the p(x)- Laplacian, Math. Nachr. 284 (2011), no. 5-6, 639-652.
  12. G. Bonanno and A. Chinni, Existence and multiplicity of weak solutions for elliptic Dirichlet problems with variable exponent, J. Math. Anal. Appl. 418 (2014), no. 2, 812-827.
  13. G. Bonanno, G. D'Agui, and P. Winkert, Sturm-Liouville equations involving discontinuous nonlinearities, Minimax Theory Appl. 1 (2016), no. 1, 125-143.
  14. 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.
  15. L. Caffarelli, Non-local diffusions, drifts and games, in Nonlinear partial differential equations, 37-52, Abel Symp., 7, Springer, Heidelberg, 2012.
  16. F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, translated from the 2007 French original by Reinie Erne, Universitext, Springer, London, 2012.
  17. 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.
  18. G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul. 7 (2008), no. 3, 1005-1028.
  19. 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: //
  20. J.-M. Kim, Y.-H. Kim, and J. Lee, Multiplicity of small or large energy solutions for Kirchhoff-Schrodinger-Type equations involving the fractional p-Laplacian in $R^N$, Symmetry 10 (2018), 1-21.
  21. N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A 268 (2000), no. 4-6, 298-305.
  22. N. Laskin, Fractional Schrodinger equation, Phys. Rev. E (3) 66 (2002), no. 5, 056108, 7 pp.
  23. 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.
  24. S. Liu, On superlinear problems without the Ambrosetti and Rabinowitz condition, Non-linear Anal. 73 (2010), no. 3, 788-795.
  25. S. Liu and S. Li, Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica (Chin. Ser.) 46 (2003), no. 4, 625-630.
  26. 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.
  27. 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.
  28. 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, R161-R208.
  29. O. H. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations 245 (2008), no. 12, 3628-3638.
  30. R. Pei, C. Ma, and J. Zhang, Existence results for asymmetric fractional p-Laplacian problem, Math. Nachr. 290 (2017), no. 16, 2673-2683.
  31. 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.
  32. 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.
  33. R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), no. 2, 887-898.
  34. Y. Wei and X. Su, Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian, Calc. Var. Partial Differential Equations 52 (2015), no. 1-2, 95-124.
  35. 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.
  36. 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.