Nontrivial Solutions for Fractional Schrödinger Equations with Electromagnetic Fields and Critical or Supercritical Growth

• Quanqing Li ^[1] ; Jianjun Nie ^[3] ; Wenbo Wang ^[2]
  1. [1] Honghe University

     Honghe University

     China

     
  2. [2] Yunnan University

     Yunnan University

     China

     
  3. [3] North China Electric Power University

  Mostrar afiliaciones +
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 2, 2024
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we study the following fractional Schrödinger equation with electromagnetic fields and critical or supercritical growth (−)s Au + V(x)u = λ|u| p−2u + f (x, |u| 2)u, x ∈ RN , where (−)s A is the fractional magnetic operator with 0 < s < 1, N > 2s, 2∗ s = 2N N−2s , λ > 0, V ∈ C(RN , R) and A ∈ C(RN , RN ) are the electric and magnetic potentials, respectively. When V and f are asymptotically periodic in x, and f is a continuous function and there exists 2 < q < 2∗ s such that | f (x, t)| ≤ C(1+|t| q−2 2 ) for all (x, t), for 2∗ s ≤ p < 22∗ s − q. For any D > 0 fixed, if λ ∈ (0, D] we prove that the equation has a nontrivial solution by the truncation method. Our method can provide a prior L∞-estimate.

• Referencias bibliográficas
  • 1. Antonelli, P., Athanassoulis, A., Hajaiej, H., Markowich, P.: On the XFEL Schrödinger equation: highly oscillatory magnetic potentials...
  • 2. Ambrosio, V., d’Avenia, P.: Nonlinear fractional magnetic Schrödinger equation: existence and multiplicity. J. Differ. Equ. 264, 3336–3368...
  • 3. Alves, C.O., Miyagaki, O.H.: Existence and concentration of solution for a class of fractional elliptic equation in RN via penalization...
  • 4. Bertoin, J.: Lévy Processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996)
  • 5. Bonheure, D., Nys, M., Van Schaftingen, J.: Properties of ground states of nonlinear Schrödinger equations under a weak constant magnetic...
  • 6. Cardoso, J., dos Prazeres, D., Severo, U.: Fractional Schrödinger equations involving potential vanishing at infinity and super-critical...
  • 7. Cheng, M.: Bound state for the fractional Schrödinger equation with unbounded potential. J. Math. Phys. 53, 043507 (2012)
  • 8. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32, 1245–1260 (2007)
  • 9. Cosmo, J., Schaftingen, J.: Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field....
  • 10. Dipierro, S., Palatucci, G., Valdinoci, E.: Existence and symmetry results for a Schrödinger type problem involving the fractional Lapacian....
  • 11. Ding, Y., Wang, Z.: Bound states of nonlinear Schrödinger equations with magnetic fields. Ann. Mat. 190, 427–451 (2011)
  • 12. d’Avenia, P., Squassina, M.: Ground states for fractional magnetic operators. ESAIM Control Optim. Calc. Var. 24, 1–24 (2018)
  • 13. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
  • 14. Felmer, P., Quaas, A., Tan, J.: Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian. Proc. R. Soc. Edinb....
  • 15. Fournais, S., Treust, L.L., Raymond, N., Van Schaftingen, J.: Semiclassical Sobolev constants for the electro-magnetic Robin Laplacian....
  • 16. Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E. 66, 056108 (2002)
  • 17. Laskin, N.: Fractional quantum mechanics and Levy path integrals. Phys. Lett. A. 268, 298–305 (2000)
  • 18. Li, Q., Teng, K., Wu, X.: Existence of positive solutions for a class of critical fractional Schrödinger equations with potential vanishing...
  • 19. Li, Q., Teng, K., Wu, X.: Ground states for Kirchhoff-type equations with critical or supercritical growth. Math. Methods Appl. Sci. 40,...
  • 20. Li, Q., Teng, K., Wu, X.: Ground states for fractional Schrödinger equations with critical growth. J. Math. Phys. 59, 033504 (2018)
  • 21. Li, Q., Teng, K., Wu, X., Wang, W.: Existence of nontrivial solutions for fractional Schrödinger equations with critical or supercritical...
  • 22. Li, Q., Wu, X.: Soliton solutions for fractional Schrödinger equations. Appl. Math. Lett. 53, 119–124 (2016)
  • 23. Liang, S., Repovs, D., Zhang, B.: On the fractional Schrödinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity....
  • 24. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, London (1978)
  • 25. Secchi, S.: Ground state solutions for nonlinear fractional Schrödinger equations in RN . J. Math. Phys. 54, 031501 (2013)
  • 26. Squasssina, M., Volzone, B.: Bourgain–Brézis–Mironescu formula for magnetic operators. C. R. Math. 354, 825–831 (2016)
  • 27. Szulkin, A., Weth, T.: The method of Nehari manifold. In: Gao, D.Y., Motreanu, D. (eds.) Handbook of Nonconvex Analysis and Applications,...
  • 28. Shang, X., Zhang, J.: Ground states for fractional Schrödinger equations with critical growth. Nonlinearity 27, 187–207 (2014)
  • 29. Wang, W., Zhou, J., Li, Y., Li, Q.: Existence of positive solutions for fractional Schrödinger–Poisson system with critical or supercritical...
  • 30. Zhang, H., Xu, J., Zhang, F.: Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in RN . J. Math....