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Multiplicity of solutions for the noncooperative Schrödinger–Kirchhoff system involving the fractional p-Laplacian in \({\mathbb {R}}^N\)

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Abstract

In this paper, we investigate the existence of solutions for the noncooperative Schrödinger–Kirchhoff-type system involving the fractional p-Laplacian and critical nonlinearities in \({\mathbb {R}}^{N}\). By applying the Limit Index Theory due to Li (Nonlinear Anal 25:1371–1389, 1995) and the fractional version of concentration-compactness principle, we obtain the existence and multiplicity of solutions for the above systems under some suitable assumptions. To our best knowledge, it seems that this is the first time to exploit the existence of solutions for the noncooperative Schrödinger–Kirchhoff-type system involving the fractional p-Laplacian and critical nonlinearity in \({\mathbb {R}}^{N}\).

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References

  1. Adams, R.: Sobolev Spaces. Academic Press, Cambridge (1975)

    MATH  Google Scholar 

  2. Alves, C.O., Corrês, F.J.S.A., Ma, T.F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Autuori, G., Fiscella, A., Pucci, P.: Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal. 125, 699–714 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benci, V.: On critical point theory for indefinite functionals in presence of symmetries. Trans. Am. Math. Soc. 274, 533–572 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bisci, G.M., Rădulescu, V., Servadei, R.: Variational methods for nonlocal fractional equations. In: Encyclopedia of Mathematics and its Applications, vol. 162. Cambridge University Press, Cambridge (2016)

  7. Brézis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  8. Caffarelli, L.A.: Non-local diffusions, drifts and games. In: Nonlinear Partial Differential Equations, Abel Symposium, vol. 7, pp. 37–52 (2012)

  9. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhike’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fiscella, A.: Infinitely many solutions for a critical Kirchhoff type problem involving a fractional operator. Differ. Integral Equ. 29, 513–530 (2016)

    MathSciNet  MATH  Google Scholar 

  11. Fiscella, A., Valdinoci, E.: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fiscella, A., Bisci, G.M., Servadei, R.: Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems. Bull. Sci. Math. 140, 14–35 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Figueiredo, G.M., Bisci, G.M., Servadei, R.: On a fractional Kirchhoff-type equation via Krasnoselskii’s genus. Asymptot. Anal. 94, 347–361 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Huang, D.W., Li, Y.Q.: Multiplicity of solutions for a noncooperative \(p\)-Laplacian elliptic system in \({\mathbb{R}}^N\). J. Differ. Equ. 215, 206–223 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Krawcewicz, W., Marzantowicz, W.: Some remarks on the Lusternik-Schnirelman method for non-differentiable functionals invariant with respect to a finite group action. Rocky Mt. J. Math. 20, 1041–1049 (1990)

    Article  MATH  Google Scholar 

  17. Li, Y.Q.: A limit index theory and its application. Nonlinear Anal. 25, 1371–1389 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lin, F., Li, Y.Q.: Multiplicity of solutions for a noncooperative elliptic system with critical Sobolev exponent. Z. Angew. Math. Phys. 60, 402–415 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lions, P.L.: The concentration-compactness principle in the calculus of variations, the limit case (I). Revista Matematica Iberoamericana 1, 145–201 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lions, P.L.: The concentration-compactness principle in the calculus of variations, the limit case (II). Revista Matematica Iberoamericana 1, 45–121 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mingqi, X., Bisci, G.M., Tian, G., Zhang, B.: Infinitely many solutions for the stationary Kirchhoff problems involving the fractional \(p\)-Laplacian. Nonlinearity 29, 357–374 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pucci, P., Saldi, S.: Critical stationary Kirchhoff equations in \({\mathbb{R}}^N\) involving nonlocal operators. Revista Matematica Iberoamericana 32, 1–22 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. 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)

    Article  MATH  Google Scholar 

  25. 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)

    MathSciNet  MATH  Google Scholar 

  26. Rabinowitz, P.H.: Minimax methods in critical-point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, vol. 65. American Mathematical Society, Providence, RI (1986)

  27. Servadei, R.: Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity. Contemp. Math. 595, 317–340 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Willem, M.: Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24. Birkhäser, Boston/Basel/Berlin (1996)

    Google Scholar 

  29. Xiang, M.Q., Zhang, B.L., Rădulescu, V.: Existence of solutions for perturbed fractional \(p\)-Laplacian equations. J. Differ. Equ. 260, 1392–1413 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Xiang, M.Q., Zhang, B.L., Rădulescu, V.: Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional \(p\)-Laplacian. Nonlinearity 29, 3186–3205 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. 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)

    Article  MathSciNet  MATH  Google Scholar 

  32. Xiang, M.Q., Zhang, B.L., Guo, X.Y.: Infinitely many solutions for a fractional Kirchhoff type problem via fountain theorem. Nonlinear Anal. 120, 299–313 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. 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. doi:10.1515/ans-2016-6002

  34. Zhang, B.L., Bisci, G.M., Servadei, R.: Superlinear nonlocal fractional problems with infinitely many solutions. Nonlinearity 28, 2247–2264 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics, Changchun Normal University, Changchun, 130032, Jilin, People’s Republic of China

    Sihua Liang

  2. School of Mathematical Science, Nanjing Normal University, Nanjing, 210046, Jiangsu, People’s Republic of China

    Jihui Zhang

Authors
  1. Sihua Liang
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  2. Jihui Zhang
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Correspondence to Jihui Zhang.

Additional information

S. Liang is 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). J.H. Zhang is supported by NSFC (No. 11571176).

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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). https://doi.org/10.1007/s00033-017-0805-9

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  • DOI: https://doi.org/10.1007/s00033-017-0805-9

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