Advertisement

Vector solutions with clustered peaks for nonlinear fractional Schrödinger systems in \(\mathbb {R}^{N}\)

  • Weiming LiuEmail author
  • Miaomiao Niu
  • Yanfang Peng
Article
  • 122 Downloads

Abstract

We consider the fractional nonlinear Schrödinger system
$$\begin{aligned} \left\{ \begin{array}{ll} \epsilon ^{2s}(-\Delta )^s u +P_1( x)u=\mu _1 |u|^{2p-2}u+\beta |v|^p|u|^{p-2}u, \quad x\in \mathbb {R}^N,\\ \epsilon ^{2s}(-\Delta )^s v +P_2( x)v=\mu _2 |v|^{2p-2}v+\beta |u|^p|v|^{p-2}v, \quad \; x\in \mathbb {R}^N,\\ \end{array} \right. \end{aligned}$$
where \(\epsilon >0\) is a small parameter, \(0<s<1,\) \(P_1\) and \(P_2\) are positive potentials, \(\mu _1>0,~\mu _2>0\), and \(\beta \in \mathbb {R}\) is a coupling constant. To construct solutions to this system, we use the Lyapunov–Schmidt reduction that takes advantage of the variational structure of the problem. For any positive integer \(k\ge 2\), we construct k interacting spikes concentrating near the local maximum point \(x_{0}\) of \(P_1\) and \(P_2\) when \(P_{1}(x_{0})=P_{2}(x_{0})\) in the attractive case. For any two positive integers \(k,m\ge 2\), we construct k interacting spikes for u near the local maximum point \(x_{1,0}\) of \(P_1\) and m interacting spikes for v near the local maximum point \(x_{2,0}\) of \(P_2\), respectively, when \(x_{1,0}\ne x_{2,0}\). For \(s = 1\), this corresponds to the system studied by Peng and Pi (Discrete Contin Dyn Syst 36:2205–2227, 2016) for the classical nonlinear Schrödinger system.

Keywords

Nonlinear fractional Schrödinger system Reduction method Vector solutions 

Mathematics Subject Classification

35B99 35J20 35J65 

References

  1. 1.
    Ambrosio, V.: Multiplicity of solutions for fractional Schrödinger systems in \(\mathbb{R}^N\), arXiv:1703.04370v1
  2. 2.
    Alves, C., Miyagaki, O.: Existence and concentration of solution for a class of fractional elliptic equation in \({\mathbb{R}}^N\) via penalization method. Calc. Var. Partial Differ. Equ. 55, 1–19 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bertoin, J.: Lévy Processes, Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996)Google Scholar
  4. 4.
    Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, vol. 20. Springer, Cham; Unione Matematica Italiana, Bologna (2016)Google Scholar
  5. 5.
    Chen, W., Deng, S.: The Nehari manifold for a fractional p-Laplacian system involving concave–convex nonlinearities. Nonlinear Anal. Real World Appl. 27, 80–92 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Capella, A., Dávila, J., Dupaigne, L., Sire, Y.: Regularity of radial extremal solutions for some nonlocal semilinear equations. Commun. Partial Differ. Eqs. 36, 1353–1384 (2011)CrossRefzbMATHGoogle Scholar
  7. 7.
    Cao, D., Peng, S.: Semi-classical bound states for Schrödinger equations with potentials vanishing or unbounded at infinity. Commun. Partial Differ. Eqs. 34, 1566–1591 (2009)CrossRefzbMATHGoogle Scholar
  8. 8.
    Colorado, E., de Pablo, A., Sánchez, U.: Perturbations of a critical fractional equation. Pac. J. Math. 271, 65–85 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Eqs. 32, 1245–260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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
  11. 11.
    Dávila, J., Del Pino, M., Wei, J.: Concentrating standing waves for the fractional nonlinear Schrödinger equation. J. Differ. Eqs. 256, 858–892 (2014)CrossRefzbMATHGoogle Scholar
  12. 12.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dipierro, S., Pinamonti, A.: A geometric inequality and a symmetry result for elliptic systems involving the fractional Laplacian. J. Differ. Eqs. 255, 85–119 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Frank, R.L., Lenzmann, E.: Uniqueness and nondegeneracy of ground states for \((-\Delta )^{s}Q + Q - Q^{\alpha +1} = 0\) in \({\mathbb{R}}\). Acta Math. 210, 261–318 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Frank, R.L., Lenzmann, E., Silvestre, L.: Uniqueness of radial solutions for the fractional Laplacian. Commun. Pure Appl. Math.  https://doi.org/10.1002/cpa.21591
  16. 16.
    Fall, M., Mahmoudi, F., Valdinoci, E.: Ground states and concentration phenomena for the fractional Schrödinger equation. Nonlinearity 28, 1937–1961 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    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)CrossRefzbMATHGoogle Scholar
  18. 18.
    Guo, Q., He, X.: Least energy solutions for a weakly coupled fractional Schrödinger system. Nonlinear Anal. 132, 141–159 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    He, Q., Peng, S., Peng, Y.: Existence, non-degeneracy of proportional positive solutions and least energy solutions for a fractional elliptic system. Adv. Differ. Eqs. 22, 867–892 (2017)MathSciNetzbMATHGoogle Scholar
  20. 20.
    He, X., Squassina, M., Zou, W.: The Nehari manifold for fractional systems involving critical nonlinearities. Commun. Pure Appl. Anal. 15, 1285–1308 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    He, X., Zou, W.: Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities. Calc. Var. Partial Differ. Eqs. 55, 1–39 (2016)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kang, X., Wei, J.: On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Adv. Differ. Eqs. 5, 899–928 (2000)zbMATHGoogle Scholar
  23. 23.
    Liu, W.: Multi-peak positive solutions for nonlinear fractional Schrödinger systems in \({\mathbb{R}}^N\). Adv. Nonlinear Stud. 2, 231–247 (2016)zbMATHGoogle Scholar
  24. 24.
    Li, Y., Ma, P.: Symmetry of solutions for a fractional system. Sci. China Math. 60, 1805–1824 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lin, F., Ni, W., Wei, J.: on the number of interior peak solutions for a singularly perturbed Neumann problem. Commun. Pure Appl. Math. 60, 252–281 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Noussair, E.S., Yan, S.: On positive multipeak solutions of a nonlinear elliptic problem. J. Lond. Math. Soc. 62, 213–227 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Peng, S., Pi, H.: Spike vector solutions for some coupled nonlinear Schrödinger equations. Discrete Contin. Dyn. Syst. 36, 2205–2227 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Peng, S., Wang, Z.: Segregated and synchronized vector solutions for nonlinear Schrödinger systems. Arch. Ration. Mech. Anal. 208, 305–339 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Quaas, A., Xia, A.: Existence results of positive solutions for nonlinear cooperative elliptic systems involving fractional Laplacian. Commun. Contemp. Math.  https://doi.org/10.1142/S0219199717500328
  30. 30.
    Secchi, S.: Ground state solutions for nonlinear fractional Schrödinger equations in \({\mathbb{R}}^N\). J. Math. Phys. 54, 031501 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    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
  32. 32.
    Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101, 275–302 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Servadei, R., Valdinoci, E.: The Brezis–Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367, 67–102 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Wang, Q.: Positive least energy solutions of fractional Laplacian systems with critical exponent. Electron. J. Differ. Eqs. 2016(150), 1–16 (2016)Google Scholar
  35. 35.
    Wei, Y., Su, X.: Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian. Calc. Var. Partial Differ. Eqs. 52, 95–124 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHubei Normal UniversityHuangshiPeople’s Republic of China
  2. 2.College of Applied ScienceBeijing University of TechnologyBeijingPeople’s Republic of China
  3. 3.Department of Mathematics and Computer ScienceGuizhou Normal UniversityGuiyangPeople’s Republic of China

Personalised recommendations