This paper investigates the existence of normalized solutions to a fractional Schr?dinger equation with combined nonlinearities. In previous studies, the equation -Δu γu=g(u) ❘u❘q-2u. where N≥3,2<q<2*=2N/N-2 has been proven to have solutions through various constraints and methods. Furthermore, we consider the existence of solutions for fractional equations. In L2 the supercritical cases, we employ the Sobolev subcritical approximation method to establish the existence of normalized ground-state solutions.
Cite this paper
Zhang, Y. (2025). Normalized Ground State Solutions to Fractional Schr?dinger Equations with?General Nonlinearities. Open Access Library Journal, 12, e2925. doi: http://dx.doi.org/10.4236/oalib.1112925.
Di Nezza, E., Palatucci, G. and Valdinoci, E. (2012) Hitchhiker's Guide to the Fractional Sobolev Spaces. Bulletin des Sciences Mathématiques, 136, 521-573. https://doi.org/10.1016/j.bulsci.2011.12.004
Silvestre, L. (2006) Regu-larity of the Obstacle Problem for a Fractional Power of the Laplace Operator. Communications on Pure and Applied Mathematics, 60, 67-112. https://doi.org/10.1002/cpa.20153
Devillanova, G. and Marano, G.C. (2016) A Free Fractional Viscous Oscillator as a Forced Standard Damped Vi-bration. Fractional Calculus and Applied Analysis, 19, 319-356. https://doi.org/10.1515/fca-2016-0018
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
Jeanjean, L. (1997) Existence of So-lutions with Prescribed Norm for Semilinear Elliptic Equations. Nonlinear Anal-ysis: Theory, Methods & Applications, 28, 1633-1659. https://doi.org/10.1016/s0362-546x(96)00021-1
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
Jeanjean, L. and Lu, S. (2019) Nonradial Normalized Solutions for Nonlinear Scalar Field Equations. Nonlinearity, 32, 4942-4966. https://doi.org/10.1088/1361-6544/ab435e
Guo, Y., Luo, Y. and Zhang, Q. (2018) Minimizers of Mass Critical Hartree Energy Functionals in Bounded Domains. Journal of Differential Equations, 265, 5177-5211. https://doi.org/10.1016/j.jde.2018.06.032
Noris, B., Tavares, H. and Verzini, G. (2019) Normalized Solutions for Nonlinear Schrödinger Systems on Bounded Domains. Nonlinearity, 32, 1044-1072. https://doi.org/10.1088/1361-6544/aaf2e0
Soave, N. (2020) Normal-ized Ground States for the NLS Equation with Combined Nonlinearities. Journal of Differential Equations, 269, 6941-6987. https://doi.org/10.1016/j.jde.2020.05.016
Zhang, P. and Han, Z. (2022) Normalized Solutions to a Kind of Fractional Schrödinger Equation with a Criti-cal Nonlinearity. Zeitschrift für angewandte Mathematik und Physik, 73, Article No. 149. https://doi.org/10.1007/s00033-022-01792-y
Ding, Y. and Zhong, X. (2022) Normalized Solution to the Schrödinger Equation with Poten-tial and General Nonlinear Term: Mass Super-Critical Case. Journal of Differen-tial Equations, 334, 194-215. https://doi.org/10.1016/j.jde.2022.06.013
Guo, L. and Li, Q. (2021) Multiple High Energy Solutions for Fractional Schrödinger Equation with Critical Growth. Calculus of Variations and Partial Differential Equations, 61, Article No. 15. https://doi.org/10.1007/s00526-021-02122-2
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
Felmer, P. and Torres, C. (2014) Radial Symmetry of Ground States for a Regional Fractional Nonlinear Schrödinger Equation. Communications on Pure and Applied Analysis, 13, 2395-2406. https://doi.org/10.3934/cpaa.2014.13.2395
Ye, F., Yu, S. and Tang, C. (2024) Limit Profiles and the Existence of Bound-States in Exterior Domains for Fractional Choquard Equations with Critical Exponent. Advances in Nonlinear Analysis, 13, Article ID: 20240020. https://doi.org/10.1515/anona-2024-0020
Sickel, W. and Skrzypczak, L. (2000) Radial Subspaces of Besov and Lizorkin-Triebel Classes: Extended Strauss Lemma and Compactness of Embeddings. The Journal of Fourier Analy-sis and Applications, 6, 639-662. https://doi.org/10.1007/bf02510700
Berestycki, H. and Lions, P. (1983) Nonlinear Scalar Field Equations, I Existence of a Ground State. Archive for Ra-tional Mechanics and Analysis, 82, 313-345. https://doi.org/10.1007/bf00250555
Shuai, W. and Wang, Q. (2015) Existence and Asymptotic Behavior of Sign-Changing Solutions for the Nonlinear Schrödinger-Poisson System in . Zeitschrift für angewandte Mathematik und Physik, 66, 3267-3282. https://doi.org/10.1007/s00033-015-0571-5