Normalized Solutions to Biharmonic Equation with Bounded Potential and Mixed Power Nonlinearities

• Senli Liu ^[1] ; Shibo Li ^[1] ; Ting Liu ^[2]
  1. [1] Hunan University of Science and Technology

     Hunan University of Science and Technology

     China

     
  2. [2] China University of Geosciences

     China University of Geosciences

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 3, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This paper is concerned with the Cauchy problem for a class of biharmonic equation with second-order dispersion term. We prove the existence and orbital stability of normalized standing waves for such equation with bounded potential and mixed power nonlinearity. Moreover, the mass collapse behavior of solutions, the ratio of energy to mass and the precise scope of Lagrange multiplier are considered as well.

• Referencias bibliográficas
  • 1. Bartsch, T., Soave, N.: A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems. J. Funct....
  • 2. Bellazzini, J., Visciglia, N.: On the orbital stability for a class of nonautonomous NLS. Indiana Univ. Math. J. 59, 1211–1230 (2010)....
  • 3. Bellazzini, J., Boussaïd, N., Jeanjean, L., Visciglia, N.: Existence and stability of standing waves for supercritical NLS with a partial...
  • 4. Ben-Artzi, M., Koch, H., Saut, J.: Dispersion estimates for fourth order Schrödinger equations, C. R. Acad. Sci. Paris Sér. I Math. 330,...
  • 5. Bonheure, D., Casteras, J., Gou, T., Jeanjean, L.: Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass...
  • 6. Bonheure, D., Casteras, J., dos Santos, E., Nascimento, R.: Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger...
  • 7. Boulenger, T., Lenzmann, E.: Blowup for biharmonic NLS. Ann. Sci. Éc. Norm. Supér. (4) 50, 503–544 (2017). https://doi.org/10.24033/asens.2326
  • 8. Boussaïd, N., Fernández, A., Jeanjean, L.: Some remarks on a minimization problem associated to a fourth order nonlinear Schrödinger equation,...
  • 9. Brézis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88,...
  • 10. Cazenave, T., Lions, P.: Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561....
  • 11. D’Avenia, P., Pomponio, A., Schino, J.: Radial and non-radial multiple solutions to a general mixed dispersion NLS equation. Nonlinearity....
  • 12. Dinh, V.: On the instability of standing waves for 3D dipolar Bose-Einstein condensates. Phys. D. 419, 132856, 12 (2021). https://doi.org/10.1016/j.physd.2021.132856
  • 13. Dinh, V.: The 3D Nonlinear Schrödinger Equation with a Constant Magnetic Field. J. Dynam. Differential Equations. (2022). https://doi.org/10.1007/s10884-022-10235-1
  • 14. Feng, B.: On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities. J. Evol. Equ. 18, 203–220...
  • 15. Feng, Z., Su, Y.: Ground state solution to the biharmonic equation. Z. Angew. Math. Phys. 73, 15, 24 (2022). https://doi.org/10.1007/s00033-021-01643-2
  • 16. Fernández, A., Jeanjean, L., Mandel, R., Mari¸s, M.: Non-homogeneous Gagliardo-Nirenberg inequalities in RN and application to a biharmonic...
  • 17. Fibich, G., Ilan, B., Papanicolaou, G.: Self-focusing with fourth-order dispersion. SIAM J. Appl. Math. 62, 1437–1462 (2002). https://doi.org/10.1137/S0036139901387241
  • 18. Guo, Q.: Scattering for the focusing L2-supercritical and H˙ 2-subcritical biharmonic NLS equations. Comm. Partial Differential Equations....
  • 19. Jeanjean, L.: Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal. 28, 1633–1659 (1997). https://doi.org/10.1016/S0362-546X(96)00021-1
  • 20. Jeanjean, L., Lu, S.: A mass supercritical problem revisited. Calc. Var. Partial Differential Equations. 59, 174 (2020). https://doi.org/10.1007/s00526-020-01828-z
  • 21. Ji, C., Su, N.: Existence and stability of standing waves for the mixed dispersion nonlinear Schrödinger equation with a partial confinement...
  • 22. Karpman, V.: Stabilization of soliton instabilities by higher order dispersion: KdV-type equations. Phys. Lett. A. 210, 77–84 (1996)....
  • 23. Karpman, V., Shagalov, A.: Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion. Phys....
  • 24. Hayek, S.: Advanced mathematical methods in science and engineering, Marcel Dekker, Inc., New York, (2001), 273-295. https://doi.org/10.1007/s00032-014-0227-5
  • 25. Lazer, A., McKenna, P.: Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM...
  • 26. Luo, T., Zheng, S., Zhu, S.: The existence and stability of normalized solutions for a bi-harmonic nonlinear Schrödinger equation with...
  • 27. Luo, X., Yang, T.: Normalized solutions for a fourth-order Schrödinger equation with a positive secondorder dispersion coefficient. Sci....
  • 28. Pausader, B.: Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case. Dyn. Partial Differ. Equ....
  • 29. Pausader, B.: The focusing energy-critical fourth-order Schrödinger equation with radial data. Discrete Contin. Dyn. Syst. 24, 1275–1292...
  • 30. Phan, T.: Blowup for biharmonic Schrödinger equation with critical nonlinearity. Z. Angew. Math. Phys. 69, 31, 11 (2018). https://doi.org/10.1007/s00033-018-0922-0
  • 31. Thompson, I., Abrahams, I.: Diffraction of flexural waves by cracks in orthotropic thin elastic plates. I. Formal solution, Proc. R. Soc....
  • 32. Yao, S., Chen, H., Radulescu, V., Sun, J.: Normalized solutions for lower critical Choquard equations with critical Sobolev perturbations....