Orbital Stability of Standing Waves for the Sobolev Critical Schrödinger Equation with Inverse-Power Potential

• Leijin Cao ^[1] ; Binhua Feng ^[1] ; Yichun Mo ^[2]
  1. [1] Northwest Normal University

     Northwest Normal University

     China

     
  2. [2] Lanzhou Jiaotong University

     Lanzhou Jiaotong University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 4, 2024
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we study the Cauchy problem for the nonlinear Schrödinger equation with focusing inverse-power potential and the Sobolev critical nonlinearity. By considering the corresponding local minimization problem, we show that the existence of ground state solutions. Then, we prove that the solution of this equation exists globally when the initial data ϕ sufficiently close to the ground states. Based on these results, we show that the set of ground states is orbitally stable.

• Referencias bibliográficas
  • 1. Bensouilah, A., Dinh, V.D., Zhu, S.: On stability and instability of standing waves for the nonlinear Schrödinger equation with an inverse-square...
  • 2. Brézis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88,...
  • 3. Cazenave, T.: Semilinear Schrödinger Equations. American Mathematics Society, New York (2003)
  • 4. Cazenave, T., Lions, P.L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Comm. Math. Phys. 85, 549–561...
  • 5. Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Global well-posedness and scattering for the energy-critical nonlinear...
  • 6. Dinh, V.D.: Global existence and blowup for a class of the focusing nonlinear Schrödinger equation with inverse-square potential. J. Math....
  • 7. Dinh, V.D.: On nonlinear Schrödinger equations with attractive inverse-power potentials. Topol. Methods Nonlinear Anal. 57, 489–523 (2021)
  • 8. Feng, B., Zhang, H.: Stability of standing waves for the fractional Schrödinger-Hartree equation. J. Math. Anal. Appl. 460, 352–364 (2018)
  • 9. Fukaya, N.: Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential....
  • 10. Fukaya, N., Ohta, M.: Strong instability of standing waves for the nonlinear Schrödinger equations with attractive inverse power potential....
  • 11. Fukuizumi, R., Ohta, M.: Stability of standing waves for nonlinear Schrödinger equations with potentials. Differential Integral Equations...
  • 12. Jeanjean, L.: Existence of solutions with prescribed norm for semilinear elliplic equations. Nonlinear Anal. 28, 1633–1659 (1997)
  • 13. Jeanjean, L., Jendrej, J., Le, T.T., Visciglia, N.: Orbital stability of ground states for a Sobolev critical Schrödinger equation. J....
  • 14. Jia, H., Luo, X.: Prescribed mass standing waves for energy critical Hartree equations. Calc. Var. 62, 71 (2023)
  • 15. Keel, M., Tao, T.: Endpoint Strichartz estimates. Amer. J. Math. 120, 955–980 (1998)
  • 16. Kenig, C.E., Merle, F.: Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation...
  • 17. Li, J., Ma, L.: Extremals to new Gagliardo-Nirenberg inequality and ground states. Appl. Math. Letters 120, 107266 (2021)
  • 18. Li, X., Zhao, J.: Orbital stability of standing waves for Schrödinger type equations with slowly decaying linear potential. Comput. Math....
  • 19. Lu, J., Miao, C., Murphy, J.: Scattering in H1 for the intercritical NLS with an inverse-square potential. J. Differential Equations 264,...
  • 20. Meng, Y.: Existence of stable standing waves for the nonlinear Schrödinger equation with attractive inverse-power potentials. AIMS Mathematics...
  • 21. Messiah, A.: Quantum Mechanics. North Holland, Amsterdam (1961)
  • 22. Ohta, M.: Stability and instability of standing waves for one-dimensional nonlinear Schrödinger equations with double power nonlinearity....
  • 23. Okazawa, N., Suzuki, T., Yokota, T.: Energy methods for abstract nonlinear Schrödinger equations. Evol. Equ. Control Theory 1, 337–354...
  • 24. Ozawa, T.: Remarks on proofs of conservation laws for nonlinear Schrödinger equations. Calc. Var. Partial Differential Equations 25, 403–408...
  • 25. Series, G.: Spectrum of Atomic Hydrogen. Oxford University Press, Oxford (1957)
  • 26. Zhang, J., Zhu, S.: Stability of standing waves for the nonlinear fractional Schrödinger equation. J. Dynam. Differential Equations. 29,...
  • 27. Zhu, S.: Existence of stable standing waves for the fractional Schrödinger equations with combined nonlinearities. J. Evol. Equ. 17, 1003–1021...