, Volume 22, Issue 3, pp 873–895 | Cite as

The positive solutions to a quasi-linear problem of fractional p-Laplacian type without the Ambrosetti–Rabinowitz condition

  • Bin Ge
  • Ying-Xin Cui
  • Liang-Liang Sun
  • Massimiliano Ferrara


In this paper, we study the existence of nontrivial solution to a quasi-linear problem where \( (-\Delta )_{p}^{s} u(x)=2\lim \nolimits _{\epsilon \rightarrow 0}\int _{\mathbb {R}^N \backslash B_{\varepsilon }(X)} \frac{|u(x)-u(y)|^{p-2} (u(x)-u(y))}{| x-y | ^{N+sp}}dy, \) \( x\in \mathbb {R}^N\) is a nonlocal and nonlinear operator and \( p\in (1,\infty )\), \( s \in (0,1) \), \( \lambda \in \mathbb {R} \), \( \Omega \subset \mathbb {R}^N (N\ge 2)\) is a bounded domain which smooth boundary \(\partial \Omega \). Using the variational methods based on the critical points theory, together with truncation and comparison techniques, we show that there exists a critical value \(\lambda _{*}>0\) of the parameter, such that if \(\lambda >\lambda _{*}\), the problem \((P)_{\lambda }\) has at least two positive solutions, if \(\lambda =\lambda _{*}\), the problem \((P)_{\lambda }\) has at least one positive solution and it has no positive solution if \(\lambda \in (0,\lambda _{*})\). Finally, we show that for all \(\lambda \ge \lambda _{*}\), the problem \((P)_{\lambda }\) has a smallest positive solution.


Fractional p-Laplacian Quasi-linear Local minimizers Mountain-pass theorem Nontrivial solution 

Mathematics Subject Classification

35P15 35P30 35R11 


  1. 1.
    Appleabeaum, D.: Lévy processes-from probability to finance quantum groups. Notices Am. Math. Soc. 51, 1336–1347 (2004)Google Scholar
  2. 2.
    Caffarelli, L.: Nonlocal equations, drifts and games. In: Nonlinear Partial Differential Equations, Abel Symposia, vol. 7, pp. 37–52 (2012)Google Scholar
  3. 3.
    Metzler, R., Klafter, J.: The restaurant at the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A. 37, 161–208 (2004)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhikers guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Servadei, R., Valdinoci, E.: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33, 2105–2137 (2013)MathSciNetMATHGoogle Scholar
  7. 7.
    Iannizzotto, A., Squassina, M.: 1/2-Laplacian problems with exponential nonlinearity. J. Math. Anal. Appl. 414, 372–385 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Dier. Equ. 32, 1245–1260 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bjorland, C., Caffarelli, L.A., Figalli, A.: Non-local gradient dependent operators. Adv. Math. 230, 1859–1894 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Franzina, G., Palatucci, G.: Fractional \(p\)-eigenvalues. Riv. Mat. Univ. Parma 5, 315–328 (2014)MathSciNetMATHGoogle Scholar
  11. 11.
    Lindgren, E., Lindqvist, P.: Fractional eigenvalues. Calc. Var. 49, 795–826 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Iannizzotto, A., Squassina, M.: Weyl-type laws for fractional \(p\)-eigenvalue problems. Asymptot. Anal. 88, 233–245 (2014)MathSciNetMATHGoogle Scholar
  13. 13.
    Di Castro, A., Kuusi, T., Palatucci, G.: Local behavior of fractional \(p\)-minimizers. arXiv:1412.4722
  14. 14.
    Brasco, L., Franzina, G.: Convexity properties of Dirichlet integrals and Picone-type inequalities. arXiv:1403.0280
  15. 15.
    Iannizzotto, A., Mosconi, S., Squassina, M.: Global H\(\ddot{o}\)lder regularity for the fractional \(p\)-Laplacian. Preprint. arXiv:1411.2956
  16. 16.
    Iannizzotto, A., Mosconi, S., Squassina, M.: A note on global regularity for the weak solutions of fractional \(p\)-Laplacian equations. arXiv:1504.01006
  17. 17.
    Iannizzotto, A., Liu, S.B., Perera, K., Squassina, A.M.: Existence results for fractional \(p\)-Laplacian problems via Morse theory. Adv. Calc. Var. 9, 101–125 (2016)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Leandeo, M.D.P., Quaas, A.: Global bifurcation for fractional \(p\)-Laplacian and application. arXiv:1412.4722
  19. 19.
    Perera, K., Squassina, M., Yang, Y.: Bifurcation and multiplicity results for critical fractional \(p\)-Laplacian problems. arXiv:1407.8061
  20. 20.
    Afrouzi, G.A., Brown, K.J.: On a diffusive logistic equation. J. Math. Anal. Appl. 225, 326–339 (1998)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Ambrosetti, A., Rabinowitz, P.: Dual variational methods in critical points theory and applications. J. Funct. Anal. 14, 349–381 (1973)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Existence of multiple solutions with precise sign information for superlinear Neumann problems. Ann. Mat. Pura Appl. 188, 679–719 (2009)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Dunford, N., Schwartz, J.T.: Linear Operator. I. General Theory. Pure and Applied Mathematics, vol. 7. Viley, New York (1958)MATHGoogle Scholar
  24. 24.
    Heikkilä, S., Lakshmikantham, V.: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations. Monographs and Textbook in Pure and Applied Mathematics, vol. 181. Marcel Dekker, New York (1994)MATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Bin Ge
    • 1
  • Ying-Xin Cui
    • 1
  • Liang-Liang Sun
    • 1
  • Massimiliano Ferrara
    • 2
  1. 1.Department of Applied MathematicsHarbin Engineering UniversityHarbinPeople’s Republic of China
  2. 2.Department of Law and EconomicsUniversity Mediterranea of Reggio CalabriaPalazzo ZaniItaly

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