Skip to main content
SpringerLink
Log in

Fractional elliptic systems with noncoercive potentials

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

We establish the existence of weak solution to the following class of fractional elliptic systems

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^s u + a(x)u = F_u(x,u,v),&{} x \in \mathbb {R}^N, \\ (-\Delta )^s v + b(x)v = F_v(x,u,v),&{} x \in \mathbb {R}^N, \end{array} \right. \end{aligned}$$

where \(s \in (0,1)\), the potentials a, b are bounded from below and may change sign. The nonlinear term \(F \in C^1(\mathbb {R}^N \times \mathbb {R} ^2,\mathbb {R})\) can be asymptotically linear or superlinear at infinity. It interacts with the eigenvalues of the linearized problem. In the proofs we apply variational methods by considering both the resonant and nonresonant case. We notice that our results are new even in the local case \(s=1\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)

    Article  MathSciNet  Google Scholar 

  2. Akhmediev, N., Ankiewicz, A.: Novel soliton states and bifurcation phenomena in nonlinear fiber couplers. Phys. Rev. Lett. 70, 2395–2398 (1993)

    Article  MathSciNet  Google Scholar 

  3. 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  Google Scholar 

  4. Bartolo, P., Benci, V., Fortunato, D.: Abstract critical point theorem and applications to some nonlinear problems with strong resonance at infinity. Nonlinear Anal. 7, 981–1012 (1983)

    Article  MathSciNet  Google Scholar 

  5. Bieganowski, B.: Systems of coupled Schrödinger equations with sign-changing nonlinearities via classical Nehari manifold approach. Complex Var. Elliptic Equ. 64, 1237–1256 (2019)

    Article  MathSciNet  Google Scholar 

  6. Bisci, G.M., Radulescu, V.D., Servadei, R.: Variational Methods for Nonlocal Fractional Problems, With a Foreword by Jean Mawhin. Encyclopedia of Mathematics and its Applications, vol. 162. Cambridge University Press, Cambridge (2016)

    Book  Google Scholar 

  7. Caffarelli, L.A.: Non-local Diffusions, Drifts and Games. Nonlinear Partial Differential Equations Abel Symp, vol. 7. Springer, Heidelberg (2012)

    Google Scholar 

  8. Costa, D.G., Magalhães, C.A.: A unified approach to a class of strongly indefinite functionals. J. Differ. Equ. 125, 521–547 (1996)

    Article  MathSciNet  Google Scholar 

  9. Costa, D.G., Magalhães, C.A.: Variational elliptic problems which are nonquadratic at infinity. Nonlinear Anal. 23, 1401–1412 (1994)

    Article  MathSciNet  Google Scholar 

  10. de Souza, M.A., Araujo, Y.L.: On a class of fractional Schrödinger equations in \(\mathbb{R}^{N}\) with sign-changing potential. Appl. Anal. 97, 538–551 (2018)

    Article  MathSciNet  Google Scholar 

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

  12. Fall, M.M., Mahmoudi, F., Valdinoci, E.: Ground states and concentration phenomena for the fractional Schrödinger equation. Nonlinearity 28, 1937–1961 (2015)

    Article  MathSciNet  Google Scholar 

  13. 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)

    Article  Google Scholar 

  14. Furtado, M.F., Maia, L.A., Silva, E.A.B.: Solutions for a resonant elliptic system with coupling in \(\mathbb{R}^N\). Commun. Partial Differ. Equ. 27, 1515–1536 (2002)

    Article  Google Scholar 

  15. Furtado, M.F., Maia, L., Silva, E.A.B.: Systems with coupling in \(\mathbb{R}^N\) for a class of noncoercive potentials, dynamical systems and differential equations. Discrete Contin. Dyn. Syst. 295–304 (2003)

  16. Furtado, M.F., Silva, E.A.B., Xavier, M.S.: Multiplicity and concentration of solutions for elliptic systems with vanishing potentials. J. Differ. Equ. 249, 2377–2396 (2010)

    Article  MathSciNet  Google Scholar 

  17. Furtado, M.F., Maia, L.A., Silva, E.A.B.: On a double resonant problem in \(\mathbb{R}^N\). Differ. Integral Equ. 15, 1335–1344 (2002)

    MATH  Google Scholar 

  18. Furtado, M.F., Silva, E.A.B.: Double resonant problems which are locally non-quadratic at infinity. In: Proceedings of the USA-Chile Workshop on Nonlinear Analysis (Viña del Mar-Valparaiso, 2000), 155–171, Electronic Journal of Differential Equations, 6, Southwest Texas State Univ., San Marcos, TX (2001)

  19. Guo, Q., He, X.: Least energy solutions for a weakly coupled fractional Schrödinger system. Nonlinear Anal. 132, 141–159 (2016)

    Article  MathSciNet  Google Scholar 

  20. Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. 268, 298–305 (2000)

    Article  MathSciNet  Google Scholar 

  21. Laskin, N.: Fractional Schrödinger equation. Phys. Rev. 66, 56–108 (2002)

    MathSciNet  Google Scholar 

  22. Lü, D., Peng, S.: On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling. Discrete Contin. Dyn. Syst. 37, 3327–3352 (2017)

    Article  MathSciNet  Google Scholar 

  23. Maia, L.A., Silva, E.A.B.: On a class of coupled elliptic systems in \(\mathbb{R}^{N}\). NoDEA Nonlinear Differ. Equ. Appl. 14, 303–313 (2007)

    Article  Google Scholar 

  24. Rabinowitz, P.: On a class of nonlinear Schrödinger equations Z. Angew. Math. Phys. 43, 270–291 (1992)

    Article  MathSciNet  Google Scholar 

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

  26. Secchi, S.: Ground state solutions for nonlinear fractional Schrödinger equations in \(\mathbb{R}^{N}\). J. Math. Phys. 54, 031501 (2013)

    Article  MathSciNet  Google Scholar 

  27. Secchi, S.: On fractional Schrödinger equations in \(\mathbb{R}^{N}\) without the Ambrosetti–Rabinowitz condition. Topol. Methods Nonlinear Anal. 47, 19–41 (2016)

    MathSciNet  MATH  Google Scholar 

  28. Servadei, R., Valdinoci, E.: Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389, 887–898 (2012)

    Article  MathSciNet  Google Scholar 

  29. Silva, E.A.B.: Subharmonic solutions for subquadratic Hamiltonian systems. J. Differ. Equ. 115, 120–145 (1995)

    Article  MathSciNet  Google Scholar 

  30. Silva, E.A.B.: Existence and multiplicity of solutions for semilinear elliptic systems. NoDEA Nonlinear Differ. Equ. Appl. 1, 339–363 (1994)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Pernambuco, Recife, PE, 50670-901, Brazil

    José Carlos de Albuquerque

  2. Departamento de Matemática, Universidade de Brasília, Brasília, DF, 70910-900, Brazil

    Marcelo F. Furtado

  3. Instituto de Matemática e Estatística, Universidade Federal de Goiás, Goiânia, GO, 74001-970, Brazil

    Edcarlos D. Silva

Authors
  1. José Carlos de Albuquerque
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcelo F. Furtado
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Edcarlos D. Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcelo F. Furtado.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

José Carlos de Albuquerque is partially supported by CAPES/Brazil. Marcelo F. Furtado is partially supported by CNPq/Brazil and FAPDF/Brazil. Edcarlos D. Silva is partially supported by CNPq grants 429955/2018-9.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Albuquerque, J.C.d., Furtado, M.F. & Silva, E.D. Fractional elliptic systems with noncoercive potentials. Z. Angew. Math. Phys. 72, 187 (2021). https://doi.org/10.1007/s00033-021-01613-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-021-01613-8

Keywords

Mathematics Subject Classification

Search

Navigation