Existence and Multiplicity Solutions for the \(p\)-Fractional Schrödinger–Kirchhoff Equations with Electromagnetic Fields and Critical Nonlinearity

  • Yueqiang SongEmail author
  • Shaoyun Shi


This paper is devoted to the study of the \(p\)-fractional Schrödinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity. By using the variational methods, we obtain the existence of mountain pass solutions \(u_{\varepsilon }\) which tend to the trivial solutions as \(\varepsilon \rightarrow 0\). Moreover, we get \(m^{\ast }\) pairs of solutions for the problem in absence of magnetic effects under some extra assumptions.


Fractional Schrödinger–Kirchhoff equations Fractional magnetic operator Critical nonlinearity Variational methods 

Mathematics Subject Classification (2010)

35J10 35B99 35J60 47G20 



Y.Q. Song was supported by NSFC (No. 11301038), the Natural Science Foundation of Jilin Province (No. 20160101244JC), Research Foundation during the 13th Five-Year Plan Period of Department of Education of Jilin Province, China (JJKH20170648KJ). S.Y. Shi was supported by NSFC grant (No. 11771177), China Automobile Industry Innovation and Development Joint Fund (No. U1664257), Program for Changbaishan Scholars of Jilin Province and Program for JLU Science, Technology Innovative Research Team (No. 2017TD-20).


  1. 1.
    Applebaum, D.: Lévy processes from probability to finance quantum groups. Not. Am. Math. Soc. 51, 1336–1347 (2004) zbMATHGoogle Scholar
  2. 2.
    Autuori, G., Pucci, P.: Elliptic problems involving the fractional Laplacian in \(\mathbb{R}^{N}\). J. Differ. Equ. 255, 2340–2362 (2013) CrossRefzbMATHGoogle Scholar
  3. 3.
    Autuori, G., Fiscella, A., Pucci, P.: Stationary Kirchhoff problems involving a fractional operator and a critical nonlinearity. Nonlinear Anal. 125, 699–714 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Benci, V.: On critical point theory of indefinite functionals in the presence of symmetries. Trans. Am. Math. Soc. 274, 533–572 (1982) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brezis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486–490 (1983) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications. Lecture Notes of the Unione Matematica Italiana, vol. 20. Springer/Unione Matematica Italiana, Cham/Bologna (2016) zbMATHGoogle Scholar
  7. 7.
    d’Avenia, P., Squassina, M.: Ground states for fractional magnetic operators. ESAIM Control Optim. Calc. Var. 24, 1–24 (2018) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dávila, J., del Pino, M., Wei, J.: Concentrating standing waves for the fractional nonlinear Schrödinger equation. J. Differ. Equ. 256, 858–892 (2014) CrossRefzbMATHGoogle Scholar
  9. 9.
    Dávila, J., del Pino, M., Valdinoci, E.: Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum. Anal. PDE 8, 1165–1235 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ding, Y.H., Lin, F.H.: Solutions of perturbed Schrödinger equations with critical nonlinearity. Calc. Var. Partial Differ. Equ. 30, 231–249 (2007) CrossRefzbMATHGoogle Scholar
  12. 12.
    Dipierro, S., Palatucci, G., Valdinoci, E.: Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Matematiche 68, 201–216 (2013) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dipierro, S., Medina, M., Valdinoci Enrico, E.: Fractional Elliptic Problems with Critical Growth in the Whole of \(\mathbb{R}^{N}\). Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], vol. 15, Edizioni della Normale, Pisa (2017). viii+152 pp. CrossRefzbMATHGoogle Scholar
  14. 14.
    Figueiredo, G.M., Molica Bisci, G., Servadei, R.: On a fractional Kirchhoff-type equation via Krasnoselskii’s genus. Asymptot. Anal. 94, 347–361 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fiscella, A.: Infinitely many solutions for a critical Kirchhoff type problem involving a fractional operator. Differ. Integral Equ. 29, 513–530 (2016) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Fiscella, A., Valdinoci, E.: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ghoussoub, N., Yuan, C.: Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans. Am. Math. Soc. 352, 5703–5743 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Iannizzotto, A., Liu, S., Perera, K., Squassina, M.: Existence results for fractional \(p\)-Laplacian problems via Morse theory. Adv. Calc. Var. 9, 101–125 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268, 298–305 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66, 056108 (2002) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Li, Y.Y., Guo, Q.Q., Niu, P.C.: Global compactness results for quasilinear elliptic problems with combined critical Sobolev–Hardy terms. Nonlinear Anal. 74, 1445–1464 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Liang, S., Shi, S.: Soliton solutions to Kirchhoff type problems involving the critical growth in \(\mathbb{R}^{N}\). Nonlinear Anal. 81, 31–41 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Liang, S., Zhang, J.: Existence of solutions for Kirchhoff type problems with critical nonlinearity in \(\mathbb{R}^{3}\). Nonlinear Anal., Real World Appl. 17, 126–136 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Liang, S., Zhang, J.: Multiplicity of solutions for the noncooperative Schrödinger–Kirchhoff system involving the fractional \(p\)-Laplacian in \(\mathbb{R}^{N}\). Z. Angew. Math. Phys. 68, 63 (2017) CrossRefzbMATHGoogle Scholar
  25. 25.
    Mingqi, X., Pucci, P., Squassina, M., Zhang, B.L.: Nonlocal Schrödinger–Kirchhoff equations with external magnetic field. Discrete Contin. Dyn. Syst. 37, 503–521 (2017) zbMATHGoogle Scholar
  26. 26.
    Molica Bisci, G., Rădulescu, V.D.: Ground state solutions of scalar field fractional Schrödinger equations. Calc. Var. Partial Differ. Equ. 54, 2985–3008 (2015) CrossRefzbMATHGoogle Scholar
  27. 27.
    Palatucci, G., Pisante, A.: Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc. Var. Partial Differ. Equ. 50, 799–829 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pucci, P., Saldi, S.: Critical stationary Kirchhoff equations in \(\mathbb{R}^{N}\) involving nonlocal operators. Rev. Mat. Iberoam. 32, 1–22 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Pucci, P., Xiang, M.Q., Zhang, B.L.: Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional \(p\)-Laplacian in \({\mathbb{R}}^{N}\). Calc. Var. Partial Differ. Equ. 54, 2785–2806 (2015) CrossRefzbMATHGoogle Scholar
  30. 30.
    Pucci, P., Xiang, M.Q., Zhang, B.L.: Existence and multiplicity of entire solutions for fractional \(p\)-Kirchhoff equations. Adv. Nonlinear Anal. 5, 27–55 (2016) MathSciNetzbMATHGoogle Scholar
  31. 31.
    Rabinowitz, P.H.: Minimax Methods in Critical-Point Theory with Applications to Differential Equations. CBME Regional Conference Series in Mathematics, vol. 65. American Mathematical Society, Providence (1986) CrossRefzbMATHGoogle Scholar
  32. 32.
    Servadei, R., Valdinoci, E.: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33, 2105–2137 (2013) MathSciNetzbMATHGoogle Scholar
  34. 34.
    Souza, M.: On a class of nonhomogeneous fractional quasilinear equations in \(\mathbb{R}^{N}\) with exponential growth. Nonlinear Differ. Equ. Appl. 22, 499–511 (2015) CrossRefzbMATHGoogle Scholar
  35. 35.
    Squassina, M., Volzone, B.: Bourgain–Brézis–Mironescu formula for magnetic operators. C. R. Math. 354, 825–831 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Wang, F., Xiang, M.Q.: Multiplicity of solutions to a nonlocal Choquard equation involving fractional magnetic operators and critical exponent. Electron. J. Differ. Equ. 2016, 306 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Willem, M.: Minimax Theorems. Birkhäuser, Boston (1996) CrossRefzbMATHGoogle Scholar
  38. 38.
    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) MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Xiang, M.Q., Zhang, B.L., Ferrara, M.: Multiplicity results for the non-homogeneous fractional \(p\)-Kirchhoff equations with concave-convex nonlinearities. Proc. R. Soc. A 471, 20150034 (2015). 14 pp. MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Xiang, M.Q., Molica Bisci, G., Tian, G.H., Zhang, B.L.: Infinitely many solutions for the stationary Kirchhoff problems involving the fractional \(p\)-Laplacian. Nonlinearity 29, 357–374 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Xiang, M.Q., Zhang, B.L., Zhang, X.: A nonhomogeneous fractional \(p\)-Kirchhoff type problem involving critical exponent in \(\mathbb{R} ^{N}\). Adv. Nonlinear Stud. (2016). zbMATHGoogle Scholar
  42. 42.
    Zhang, X., Zhang, B.L., Repovš, D.: Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials. Nonlinear Anal. 142, 48–68 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Zhang, B.L., Squassina, M., Zhang, X.: Fractional NLS equations with magnetic field, critical frequency and critical growth. Manuscr. Math. 155, 115–140 (2018) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Scientific Research DepartmentChangchun Normal UniversityChangchunP.R. China
  2. 2.School of MathematicsJilin UniversityChangchunP.R. China
  3. 3.State Key Laboratory of Automotive Simulation and ControlJilin UniversityChangchunP.R. China

Personalised recommendations