Infinitely Many Solutions for Fractional p-Kirchhoff Equations



In this paper we consider the existence of infinitely many weak solutions for fractional Schrödinger–Kirchhoff problems. Precisely speaking, we investigate
$$\begin{aligned} \left\{ \begin{array}{cl} M\left( \int _{\mathbb {R}^{2n}}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\mathrm{d}x\mathrm{d}y\right) (-\triangle )_p^su+V(x)|u|^{p-2}u=f(x,u), &{}\quad \mathrm{in}~\Omega ,\\ u=0, &{}\quad \mathrm{in}~\mathbb {R}^n\setminus \Omega , \end{array}\right. \end{aligned}$$
where \(\Omega \) is a bounded subset with Lipshcitz boundary \(\partial \Omega \), \(0<s<1\) is fixed, and \(1<p<n\), \((-\triangle )_p^s\) is the fractional p-Laplacian operator. Kirchhoff function M, potential function V and nonlinearity f satisfy some suitable assumptions. Under those conditions, some new multiplicity results are obtained by applying the fountain theorem and the dual fountain theorem.


Schrödinger–Kirchhoff-type equation fractional p-Laplacian fountain theorem dual fountain theorem 

Mathematics Subject Classification

35R11 35A15 


  1. 1.
    Figueiredo, G.M., Santos Junior, J.R.: Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth. Differ. Integr. Equ. 25(9–10), 853–868 (2012)MathSciNetMATHGoogle Scholar
  2. 2.
    Figueiredo, G.M.: Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument. J. Math. Anal. Appl. 401, 706–713 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Colasuonno, F., Pucci, P.: Multiplicity of solutions for \(p(x)\)-polyharmonic Kirchhoff equations. Nonlinear Anal. 74, 5962–5974 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Autuori, G., Fiscella, A., Pucci, P.: Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal. 125, 699–714 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Servadei, R., Valdinoci, E.: The Brezis-Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367, 67–102 (2015)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, C.S., Song, H.X., Xiu, Z.H.: Multiple solution for \(p\)-Kirchhoff equations in \({\mathbb{R}}^N\). Nonlinear Anal. 86, 146–156 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Nyamoradi, N.: Existence of three solutions for Kirchhoff nonlocal operators of elliptic type. Math. Commun. 18, 489–502 (2013)MathSciNetMATHGoogle Scholar
  8. 8.
    Nyamoradi, N., Chung, N.T.: Existence of solutions to nonlocal Kirchhoff equations of elliptic type via genus theory. Electron. J. Differ. Equ. 2014, 1–12 (2014)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Nyamoradi, N., Teng, K.: Existence of solutions for a Kirchhoff-type-nonlocal operators of elliptic type. Commun. Pure Appl. Anal. 14, 361–371 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Xiang, M.Q., Bisci, G.M., Tian, G.H., Zhang, B.L.: Infinitely many solutions for the stationary Kirchhoff problems involving the fractional \(p\)-Laplacian. Nonlinearity 29, 357–374 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fiscella, A., Valdinoci, E.: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Pucci, P., Saldi, S.: Critical stationary Kirchhoff equations in \({\mathbb{R}}^N\) involving nonlocal operators. Rev. Mat. Iberoam. 32(1), 1–22 (2016)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    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)CrossRefMATHGoogle Scholar
  14. 14.
    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)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Bartsch, T., Willem, M.: On an elliptic equation with concave and convex nonlinearities. Proc. Am. Math. Soc. 123, 3555–3561 (1995)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Willem, M.: Minimax Theorems. Progress in Nonlinear Differential Equations and Their Applications, vol. 24. Birkhäuser Boston, lnc., Boston (1996)Google Scholar
  19. 19.
    Teng, K.M.: Multiple solutions for a class of fractional Schrödinger equations in \({\mathbb{R}}^N\). Nonlinear Anal. Real World Appl. 21, 76–86 (2015)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Chang, X.J.: Ground state solutions of asymptotically linear fractional Schrödinger equations. J. Math. Phys. 54(6), 061504 (2013)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Autuori, G., Pucci, P.: Elliptic problems involving the fractional Laplacian in \({\mathbb{R}}^N\). J. Differ. Equ. 255, 2340–2362 (2013)CrossRefMATHGoogle Scholar
  22. 22.
    Ge, B.: Multiple solutions of nonlinear Schrödinger equation with the fractional Laplacian. Nonlinear Anal. Real World Appl. 30, 236–247 (2016)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Bartsch, T.: Infinitely many solutions of a symmetric Dirichlet problem. Nonlinear Anal. 20, 1205–1216 (1993)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    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)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Binlin, Z., Molica Bisci, G., Servadei, R.: Superlinear nonlocal fractional problems with infinitely many solutions. Nonlinearity 28, 2247–2264 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of ScienceHohai UniversityNanjingPeople’s Republic of China
  2. 2.Faculty of Mathematics and PhysicsHuaiyin Institute of TechnologyHuai’anPeople’s Republic of China

Personalised recommendations