Advertisement

manuscripta mathematica

, Volume 158, Issue 1–2, pp 159–203 | Cite as

Multiplicity of concentrating solutions for a class of fractional Kirchhoff equation

  • Xiaoming HeEmail author
  • Wenming Zou
Article
  • 144 Downloads

Abstract

We study the multiplicity of concentrating solutions to the nonlinear fractional Kirchhoff equation
$$\begin{aligned} \left( \varepsilon ^{2s}a+\varepsilon ^{4s-3}b\int _{\mathbb R^3}|(-\Delta )^{\frac{s}{2}}u|^2dx\right) (-\Delta )^s u+V(x)u=f(u)~~\text{ in }~~\mathbb R^3, \end{aligned}$$
where \(\varepsilon >0\) is a positive parameter, \((-\Delta )^s\) is the fractional laplacian with \(s\in (\frac{3}{4},1), a,b\) are positive constants, and V is a positive potential such that \(\inf _{\partial \Lambda }V>\inf _{\Lambda }V\) for some open bounded subset \(\Lambda \subset \mathbb R^3.\) We relate the number of positive solutions with the topology of the set where V attains its minimum in \(\Lambda \). The proof is based on the Ljusternik–Schnirelmann theory.

Mathematics Subject Classification

47G20 35J50 35B65 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

We would like to thank the anonymous referees for careful reading the manuscript and suggesting many valuable comments. X. He is supported by the National Natural Science Foundation of China (Grant No. 11771468, 11271386). W. Zou is supported by the National Natural Science Foundation of China (Grant No. 11771234, 11371212).

References

  1. 1.
    Alves, C.O., Miyagaki, O.H.: Existence and concentration of solutions for a class of fractional elliptic equation in \(\mathbb{R}^N\) via penalization method. Calc. Var. Partial Differ. Equ. 55, 47 (2016).  https://doi.org/10.1007/s00526-016-0983-x CrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrosetti, A., Malchiodi, A.: Perturbation Methods and Semilinear Elliptic Problems on \({\mathbb{R}}^N\). Verlag, Birkhäuser (2006)zbMATHGoogle Scholar
  3. 3.
    Ambrosetti, A., Rabinowitz, P.H.: Dual varitional methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)CrossRefzbMATHGoogle Scholar
  4. 4.
    Ambrosio, V.: Concentrating solutions for a class of nonlinear fractional Schrödinger equations in \({\mathbb{R}}^N\). arXiv: 1612.02388vl [math.AP] (7 Dec 2016)
  5. 5.
    Ambrosio, V., Isernia, T.: A multiplicity result for a fractional Kirchhoff equation in \(\mathbb{R}^N\) with a general nonlinearity. Commun. Contemp. Math. (2016).  https://doi.org/10.1142/S0219199717500547 zbMATHGoogle Scholar
  6. 6.
    D’Ancona, P., Spagnolo, S.: Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 108, 247–262 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Arosio, A., Panizzi, S.: On the well-posedness of the Kirchhoff string. Tran. Am. Math. Soc. 348, 305–330 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Autuori, G., Fiscella, A., Pucci, P.: Stationary Kirchhoff problems involving a fractional el liptic operator and a critical nonlinearity. Nonlinear Anal. 125, 699–714 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Benci, V., Cerami, G.: Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology. Calc. Var. Partial Differ. Equ. 2, 29–48 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Barrios, B., Colorado, E., Servadei, R., Soria, F.: A critical fractional equation with concave-convex power nonlinearities. Ann. Inst. H. Poincaré Anal. Non Linéaire 32, 875–900 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    C. Bucur, E. Valdinoci, Nonlocal difusion and applications. Lecture Notes of the Unione Matematica Italiana, 20. Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. xii+155 pp. ISBN: 978-3-319-28738-6; 978-3-319-28739-3Google Scholar
  12. 12.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Partial Differ. Equ. 32, 1245–1260 (2007)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chen, G.: Multiple semiclassical standing waves for fractional nonlinear Schrödinger equations. Nonlinearity 28, 927–949 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dávila, J., del Pino, M., Dipierro, S., Valdinoci, E.: Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum. Anal. PDE 8, 1165–1235 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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
  17. 17.
    Dipierro, S., Medina, M., Peral, I., Valdinoci, E.: Bifurcation results for a fractional elliptic equation with critical exponent in \({\mathbb{R}}^n\). Manuscripta Math. 153, 183–230 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dipierro, S., Medina, M., Valdinoci, E.: Fractional elliptic problems with critical growth in the whole of \({\mathbb{R}}^n\), Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) (Lecture Notes. Scuola Normale Superiore di Pisa (New Series)), 15. Edizioni della Normale, Pisa, 2017. viii+152, pp. ISBN: 978-88-7642-600-1; 978-88-7642-601-8Google Scholar
  19. 19.
    Fall, M.M., Mahmoudi, F., Valdinoci, E.: Ground states and concentration phenomena for the fractional Schrödinger equation. Nonlinearity 28, 1937–1961 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Felmer, P., Quaas, A., Tan, J.: Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc. Roy. Soc. Edinb. 142 A, 1237–1262 (2012)CrossRefzbMATHGoogle Scholar
  21. 21.
    Figueiredo, G.M., Ikoma, N., Santos, J.R.: Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities. Arch. Ration. Mech. Anal. 213, 931–979 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Figueiredo, G.M., Santos Junior, J.R.: Multiplicity and concentration behavior of positive solutions for a Schrödinger–Kirchhoff type problem via penalization method. ESAIM: Control Opti. Calc. Var 20, 389–415 (2014)zbMATHGoogle Scholar
  23. 23.
    Figueiredo, G.M., Siciliano, G.: A multiplicity result via Ljusternik-Schnirelmann category and Morse theory for a fractional Schrödinger equation in \({\mathbb{R}}^N\), Nonlinear Differ. Equ. Appl.23 (2016).  https://doi.org/10.1007/s00030-016-0355-4
  24. 24.
    Fiscella, A., Valdinoci, E.: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156–170 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ge, B., Zhang, C.: Existence of a positive solution to Kirchhoff problems involving the fractional Laplacian. J. Anal. Appl. 34, 419–434 (2015)MathSciNetzbMATHGoogle Scholar
  26. 26.
    He, Y., Li, G., Peng, S.: Concentrating bound states for Kirchhoff type problems in \({\mathbb{R}}^3\) involving critical Sobolev exponents. Adv. Nonl. studi. 14, 483–510 (2014)zbMATHGoogle Scholar
  27. 27.
    He, X., Zou, W.: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \({\mathbb{R}}^3\). J. Differ. Equ. 252, 1813–1834 (2012)CrossRefzbMATHGoogle Scholar
  28. 28.
    He, X., Zou, W.: Ground states for nonlinear Kirchhoff equations with critical growth. Ann. Mat. Pura Appl (4) 193, 473–500 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Jin, T., Li, Y.Y., Xiong, J.: On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions. J. Eur. Math. Soc. 16, 1111–1171 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)zbMATHGoogle Scholar
  31. 31.
    Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66, 056108–056114 (2002)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Laskin, N.: Fractional quantum mechanics and Levy path integrals. Phys. Lett. A 268, 298–305 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Liang, Z., Li, F., Shi, J.: Positive solutions to Kirchhof type equations with nonlinearity having prescribed asymptotic behavior. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 155–167 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proceedings of the International Symposium Inst. Mat. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977. In: North-Holland Mathematics Studies vol. 30, Amsterdam: North-Holland, pp. 284–346 (1978)Google Scholar
  35. 35.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations, The limit case. I. Rev. Math. Iberoam 1(1), 145–201 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Lions, P.L.: The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Math. Iberoam 1(2), 45–121 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hithiker’s guide to the frctional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Perera, K., Zhang, Z.: Nontrivial solutions of Kirchhoff type problems via the Yang index. J. Differ. Equ. 221, 246–255 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    del Pino, M., Felmer, P.L.: Local mountain pass for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differ. Equ. 4, 121–137 (1996)CrossRefzbMATHGoogle Scholar
  40. 40.
    Pucci, P., Saldi, S.: Critical stationary Kirchhoff equations in \(\mathbb{R}^N\) involving nonlocal operators. Rev. Mat. Iberoam. 32, 1–22 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Pucci, P., Xiang, M., Zhang, B.: Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff equations involving the fractional \(p\)-Laplacian in \(\mathbb{R}^N,\) Calc. Var. Partial Differ. Equ. 54, 2785–2806 (2015)CrossRefzbMATHGoogle Scholar
  42. 42.
    Rabinowitz, P.H.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Shang, X., Zhang, J.: On fractional Schrödinger equation in \(\mathbb{R}^N\) with critical growth. J. Math. Phys. 54, 121502 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Secchi, S.: Ground state solutions for nonlinear fractional Schrödinger equations in \(\mathbb{R}^N\). J. Math. Phys. 54, 031501 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Servadei, R., Valdinoci, E.: The Brezis–Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367, 67–102 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discret. Contin. Dyn. Syst. 33, 2105–2137 (2013)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60, 67–112 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Szulkin, A., Weth, T.: The method of Nehari manifold. Handbook of Nonconvex Analysis and Applications, pp. 2314–2351. Inernational Press, Boston (2010)Google Scholar
  49. 49.
    Wang, J., Tian, L., Xu, J., Zhang, F.: Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J. Differ. Equ. 253, 2314–2351 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Willem, M.: Minimax Theorems. Birckhäuser, Boston (1996)CrossRefzbMATHGoogle Scholar
  51. 51.
    Wu, X.: Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff type equations in \(\mathbb{R}^N\). Nonlinear Anal. Real World Appl. 12, 1278–1287 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Zhang, X., Zhang, C.: Existence of solutions for critical fractional Kirchhoff problems. Math. Meth. Appl. Sci. 40, 1649–1665 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of ScienceMinzu University of ChinaBeijingChina
  2. 2.Department of Mathematical SciencesTsinghua UniversityBeijingChina

Personalised recommendations