|
|
Pure Mathematics 2025
带混合色散项的非线性薛定谔方程归一化驻波解的存在性
|
Abstract:
本文研究了带混合色散项的非线性薛定谔方程归一化驻波解的存在性结果，证明了三维情况下方程基态解的存在性。主要研究方法是基于极小极大方法和集中紧性原理。
In this paper, we study the existence of normalized standing wave solutions for a nonlinear Schr？dinger equation with mixed dispersion, and prove the existence of ground state solutions in the three-dimensional case. The main analytical approaches are based on the minimax approach and Concentration-compactness Lemma.
| [1] | Karpman, V.I. (1996) Stabilization of Soliton Instabilities by Higher Order Dispersion: KdV-Type Equations. Physics Letters A, 210, 77-84. https://doi.org/10.1016/0375-9601(95)00752-0 |
| [2] | Karpman, V.I. and Shagalov, A.G. (2000) Stability of Solitons Described by Nonlinear Schrödinger-Type Equations with Higher-Order Dispersion. Physica D: Nonlinear Phenomena, 144, 194-210. https://doi.org/10.1016/s0167-2789(00)00078-6 |
| [3] | Bonheure, D., Casteras, J., Gou, T. and Jeanjean, L. (2019) Normalized Solutions to the Mixed Dispersion Nonlinear Schrödinger Equation in the Mass Critical and Supercritical Regime. Transactions of the American Mathematical Society, 372, 2167-2212. https://doi.org/10.1090/tran/7769 |
| [4] | Jeanjean, L. (1997) Existence of Solutions with Prescribed Norm for Semilinear Elliptic Equations. Nonlinear Analysis: Theory, Methods & Applications, 28, 1633-1659. https://doi.org/10.1016/s0362-546x(96)00021-1 |
| [5] | Ambrosetti, A. and Malchiodi, A. (2009) Nonlinear Analysis and Semilinear Elliptic Problems. Cambridge University Press. |
| [6] | Derrick, G.H. (1964) Comments on Nonlinear Wave Equations as Models for Elementary Particles. Journal of Mathematical Physics, 5, 1252-1254. https://doi.org/10.1063/1.1704233 |
| [7] | Pohožaev, S.I. (1965) On the Eigenfunctions of the Equation. Doklady Akademii Nauk, 165, 36-39. |
| [8] | Bonheure, D., Castéras, J., Gou, T. and Jeanjean, L. (2017) Strong Instability of Ground States to a Fourth Order Schrödinger Equation. International Mathematics Research Notices, 2019, 5299-5315. https://doi.org/10.1093/imrn/rnx273 |
| [9] | d’Avenia, P., Pomponio, A. and Schino, J. (2023) Radial and Non-Radial Multiple Solutions to a General Mixed Dispersion NLS Equation. Nonlinearity, 36, 1743-1775. https://doi.org/10.1088/1361-6544/acb62d |
| [10] | Ghoussoub, N. (1993) Duality and Perturbation Methods in Critical Point Theory. Cambridge University Press. https://doi.org/10.1017/cbo9780511551703 |
| [11] | Ghoussoub, N. and Preiss, D. (1989) A General Mountain Pass Principle for Locating and Classifying Critical Points. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 6, 321-330. https://doi.org/10.1016/s0294-1449(16)30313-4 |
| [12] | Bellazzini, J., Forcella, L. and Georgiev, V. (2023) Ground State Energy Threshold and Blow-Up for NLS with Competing Nonlinearities. Annali Scuola Normale Superiore—Classe di Scienze, 24, 955-988. https://doi.org/10.2422/2036-2145.202005_044 |
| [13] | Lions, P.L. (1984) The Concentration-Compactness Principle in the Calculus of Variations. the Locally Compact Case, Part 2. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 1, 223-283. https://doi.org/10.1016/s0294-1449(16)30422-x |
