Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in \({\mathbb {R}}^N\)

  • Patrizia Pucci
  • Mingqi Xiang
  • Binlin Zhang


In this paper we investigate the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrödinger–Kirchhoff type
$$\begin{aligned} M\left( \iint _{R^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right) (-\varDelta )^s_pu+V(x)|u|^{p-2}u=f(x,u)+g(x) \end{aligned}$$
in \({\mathbb {R}}^N\), where \((-\varDelta )^s_p\) is the fractional p-Laplacian operator, with \(0<s<1<p<\infty \) and \(ps<N\), the nonlinearity \(f:{\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a Carathéodory function and satisfies the Ambrosetti–Rabinowitz condition, \(V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}^+\) is a potential function and \(g:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\) is a perturbation term. We first establish Batsch–Wang type compact embedding theorem for the fractional Sobolev spaces. Then multiplicity results are obtained by using the Ekeland variational principle and the Mountain Pass theorem.

Mathematics Subject Classification

35R11 35A15 35J60 47G20 



P. Pucci is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica “G. Severi” (INdAM) and was partially supported by the MIUR Project Aspetti variazionali e perturbativi nei problemi differenziali nonlineari, and finally the manuscript was realized within the auspices of the INDAM-GNAMPA Project 2015 titled Modelli ed equazioni non-locali di tipo frazionario (Prot_2015_000368). M. Xiang was support by the Fundamental Research Funds for the Central Universities (No. 3122015L014). B. Zhang was supported by Natural Science Foundation of Heilongjiang Province of China (No. A201306) and Research Foundation of Heilongjiang Educational Committee (No. 12541667) and Doctoral Research Foundation of Heilongjiang Institute of Technology (No. 2013BJ15).


  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, New York (2003)zbMATHGoogle Scholar
  2. 2.
    Alves, C.O., Corrês, F.J.S.A., Ma, T.F.: Positive solutions for a equasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambrosetti, A., Rabinowiz, P.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)zbMATHCrossRefGoogle Scholar
  4. 4.
    Applebaum, D.: Lévy processes—from probability to finance quantum groups. Notices Am. Math. Soc. 51, 1336–1347 (2004)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Autuori, G., Pucci, P.: Elliptic problems involving the fractional Laplacian in \({\mathbb{R}}^N\). J. Differ. Equ. 255, 2340–2362 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Barrios, B., Colorado, E., De Pablo, A., Sanchez, U.: On some critical problems for the fractional Laplacian operator. J. Differ. Equ. 252, 6133–6162 (2012)zbMATHCrossRefGoogle Scholar
  7. 7.
    Bartsch, T., Pankov, A., Wang, Z.Q.: Nonlinear Schrödinger equations with steep potential well. Commun. Contemp. Math. 3, 1–21 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bartsch, T., Wang, Z.Q.: Existence and multiplicity results for some superlinear elliptic problems on \({\mathbb{R}}^{N}\). Commun. Partial Differ. Equ. 20, 1725–1741 (1995)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Brézis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer, New York (2011)zbMATHGoogle Scholar
  10. 10.
    Caffarelli, L.: Nonlocal equations, drifts and games. Nonlinear Partial Differ. Equ. Abel Symp. 7, 37–52 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Caffarelli, L., Valdinoci, E.: Uniform estimates and limiting arguments for nonlocal minimal surfaces. Calc. Var. Partial Differ. Equ. 41, 203–240 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Chang, X., Wang, Z.Q.: Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity. Nonlinearity 26, 479–494 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Chang, X., Wang, Z.Q.: Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian. J. Differ. Equ. 256, 2965–2992 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Chen, C., Song, H., Xiu, Z.: Multiple solution for \(p\)-Kirchhoff equations in \({\mathbb{R}}^{N}\). Nonlinear Anal. 86, 146–156 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Chen, S.J., Lin, L.: Multiple solutions for the nonhomogeneous Kirchhoff equation on \({\mathbb{R}}^N\). Nonlinear Anal. RWA 14, 1477–1486 (2013)zbMATHCrossRefGoogle Scholar
  17. 17.
    Colasuonno, F., Pucci, P.: Multiplicity of solutions for \(p(x)\)-polyharmonic elliptic Kirchhoff equations. Nonlinear Anal. 74, 5962–5974 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Corrěa, F.J.S.A., Figueiredo, G.M.: On an elliptic equation of \(p\)-Kirchhoff type via variational methods. Bull. Austral. Math. Soc. 74, 236–277 (2006)Google Scholar
  19. 19.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Dipierro, S., Palatucci, G., Valdinoci, E.: Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Matematiche 68, 201–216 (2013)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Felmer, P., Quaas, A., Tan, J.: Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb. Sect. A 142, 1237–1262 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Ferrara, M., Guerrini, L., Zhang, B.L.: Multiple solutions for perturbed non-local fractional Laplacian equations. Electron. J. Differ. Equ. 2013 (2013)Google Scholar
  24. 24.
    Ferrara, M., Molica Bisci, G., Zhang, B.L.: Existence of weak solutions for non-local fractional problems via Morse theory. Discrete Contin. Dyn. Syst. Ser. B 19, 2483–2499 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Fiscella, A., Valdinoci, E.: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Franzina, G., Palatucci, G.: Fractional \(p\)-eigenvalues. Riv. Mat. Univ. Parma 5, 315–328 (2014)MathSciNetGoogle Scholar
  27. 27.
    Iannizzotto A., Liu S., Perera K., Squassina M., Existence results for fractional \(p\)-Laplacian problems via Morse theory. Adv. Calc. Var. doi: 10.1515/acv-2014-0024
  28. 28.
    Iannizzotto, A., Squassina, M.: Weyl-type laws for fractional \(p\)-eigenvalue problems. Asymptotic Anal. 88, 233–245 (2014)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)Google Scholar
  30. 30.
    Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268, 298–305 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66, 056108 (2002)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Lindgren, E., Lindqvist, P.: Fractional eigenvalues. Calc. Var. Partial Differ. Equ. 49, 795–826 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Lions, P.L.: Symétrie et compacité dans les espaces de Sobolev. J. Funct. Anal. 49, 315–334 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations, the locally compact case. Ann. Inst. H. Poincaré Anal. Non Linéaire 1, 109–145, 223–283 (1984)Google Scholar
  35. 35.
    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)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Molica Bisci, G.: Fractional equations with bounded primitive. Appl. Math. Lett. 27, 53–58 (2014)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Molica Bisci, G., Pansera, B.A.: Three weak solutions for nonlocal fractional equations. Adv. Nonlinear Stud. 14, 619–630 (2014)zbMATHMathSciNetGoogle Scholar
  38. 38.
    Molica Bisci, G., Servadei, R.: A bifurcation result for non-local fractional equations. Anal. Appl. 13, 371–394 (2015)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Nyamoradi, N.: Existence of three solutions for Kirchhoff nonlocal operators of elliptic type. Math. Commun. 18, 489–502 (2013)zbMATHMathSciNetGoogle Scholar
  40. 40.
    Pucci, P., Saldi, S.: Critical stationary Kirchhoff equations in \({\mathbb{R}}^{N}\) involving nonlocal operators. Rev. Mat. Iberoam. (2016, to appear)Google Scholar
  41. 41.
    Pucci, P., Zhang, Q.: Existence of entire solutions for a class of variable exponent elliptic equations. J. Differ. Equ. 257, 1529–1566 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Secchi, S.: Ground state solutions for nonlinear fractional Schrödinger in \({\mathbb{R}}^{N}\). J. Math. Phys. 54, 031501 (2013)Google Scholar
  43. 43.
    Servadei, R., Valdinoci, E.: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33, 2105–2137 (2013)zbMATHMathSciNetGoogle Scholar
  45. 45.
    Strauss, W.A.: Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55, 149–162 (1977)zbMATHCrossRefGoogle Scholar
  46. 46.
    Tan, J.: The Brezis–Nirenberg type problem involving the square root of the Laplacian. Calc. Var. Partial Differ. Equ. 36, 21–41 (2011)CrossRefGoogle Scholar
  47. 47.
    Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996)zbMATHCrossRefGoogle Scholar
  48. 48.
    Xiang, M.Q., Zhang, B.L., Ferrara, M.: Existence of solutions for Kirchhoff type problem involving the non-local fractional \(p\)-Laplacian. J. Math. Anal. Appl. 424, 1021–1041 (2015)zbMATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Zhang, B.L., Ferrara, M.: Multiplicity of solutions for a class of superlinear non-local fractional equations. Complex Var. Elliptic Equ. 60, 583–595 (2015)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Zhang, B.L., Ferrara, M.: Two weak solutions for perturbed non-local fractional equations. Appl. Anal. 94, 891–902 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di PerugiaPerugiaItaly
  2. 2.College of ScienceCivil Aviation University of ChinaTianjinPeople’s Republic of China
  3. 3.Chern Institute of Mathematics and LPMCNankai UniversityTianjinPeople’s Republic of China
  4. 4.Department of MathematicsHeilongjiang Institute of TechnologyHarbinPeople’s Republic of China

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