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Pure Mathematics 2025
分数阶非线性薛定谔系统规范解的存在性
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Abstract:
本文主要研究了一类分数阶非线性薛定谔系统 在 中的规范解的存在性，且解满足 其中规定 是以拉格朗日乘子出现的未知参数， 为有界连续函数。考虑
| [1] | Applebaum, D. (2004) Levy Processes—From Probability to Finance and Quantum Groups, Notices. American Mathematical Society, 51, 1336-1347. |
| [2] | Servadei, R. and Valdinoci, E. (2014) The Brezis-Nirenberg Result for the Fractional Laplacian. Transactions of the American Mathematical Society, 367, 67-102. https://doi.org/10.1090/s0002-9947-2014-05884-4 |
| [3] | Yang, Y., Zhang, J. and Shang, X. (2013) Positive Solutions of Nonhomogeneous Fractional Laplacian Problem with Critical Exponent. Communications on Pure and Applied Analysis, 13, 567-584. https://doi.org/10.3934/cpaa.2014.13.567 |
| [4] | Yan, S., Yang, J. and Yu, X. (2015) Equations Involving Fractional Laplacian Operator: Compactness and Application. Journal of Functional Analysis, 269, 47-79. https://doi.org/10.1016/j.jfa.2015.04.012 |
| [5] | 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 |
| [6] | Hirata, J. and Tanaka, K. (2019) Nonlinear Scalar Field Equations with L2 Constraint: Mountain Pass and Symmetric Mountain Pass Approaches. Advanced Nonlinear Studies, 19, 263-290. https://doi.org/10.1515/ans-2018-2039 |
| [7] | Molle, R., Riey, G. and Verzini, G. (2022) Normalized Solutions to Mass Supercritical Schrödinger Equations with Negative Potential. Journal of Differential Equations, 333, 302-331. https://doi.org/10.1016/j.jde.2022.06.012 |
| [8] | Bartsch, T. and Jeanjean, L. (2017) Normalized Solutions for Nonlinear Schrödinger Systems. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 148, 225-242. https://doi.org/10.1017/s0308210517000087 |
| [9] | Yang, T. (2020) Normalized Solutions for the Fractional Schrödinger Equation with a Focusing Nonlocal L2-Critical or L2-Supercritical Perturbation. Journal of Mathematical Physics, 61, Article 051505. https://doi.org/10.1063/1.5144695 |
| [10] | Zhang, X., Squassina, M. and Zhang, J. (2024) Multiplicity of Normalized Solutions for the Fractional Schrödinger Equation with Potentials. Mathematics, 12, Article 772. https://doi.org/10.3390/math12050772 |
| [11] | Luo, H. and Zhang, Z. (2020) Normalized Solutions to the Fractional Schrödinger Equations with Combined Nonlinearities. Calculus of Variations and Partial Differential Equations, 59, Article No. 143. https://doi.org/10.1007/s00526-020-01814-5 |
| [12] | Frank, R.L., Lenzmann, E. and Silvestre, L. (2015) Uniqueness of Radial Solutions for the Fractional Laplacian. Communications on Pure and Applied Mathematics, 69, 1671-1726. https://doi.org/10.1002/cpa.21591 |
| [13] | Fall, M.M. and Valdinoci, E. (2014) Uniqueness and Nondegeneracy of Positive Solutions of in RN When s Is Close to 1, Communications in Mathematical Physics, 69, 383-404. https://doi.org/10.1007/s00220-014-1919-y |
| [14] | Berestycki, H. and Lions, P.-L. (1983) Nonlinear Scalar Field Equations, II Existence of Infinitely Many Solutions. Archive for Rational Mechanics and Analysis, 82, 347-375. https://doi.org/10.1007/bf00250556 |
| [15] | Willem, M. (1996) Minimax Theorems. Birkhauser. |
| [16] | Brändle, C., Colorado, E., de Pablo, A. and Sánchez, U. (2013) A Concave—Convex Elliptic Problem Involving the Fractional Laplacian. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 143, 39-71. https://doi.org/10.1017/s0308210511000175 |
| [17] | Almgren, F.J. and Lieb, E.H. (1989) Symmetric Decreasing Rearrangement Is Sometimes Continuous. Journal of the American Mathematical Society, 2, 683-773. https://doi.org/10.1090/s0894-0347-1989-1002633-4 |
| [18] | Felmer, P., Quaas, A. and Tan, J. (2012) Positive Solutions of the Nonlinear Schrödinger Equation with the Fractional Laplacian. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 142, 1237-1262. https://doi.org/10.1017/s0308210511000746 |
