Normalized solutions to the fractional Schrödinger equations with potential and saturable nonlinearity
DOI:
https://doi.org/10.64700/mmm.53Keywords:
normalized solutions, Schrödinger equations, saturable nonlinearity, potential functionAbstract
This paper is concerned with the existence of normalized solutions to a kind of fractional Schrödinger equations driven by a fractional operator with a parametric potential term and a saturable nonlinear term. We achieve the minimization of the energy functional and prove the existence of normalized solutions for the equation under specific conditions that we assume for the potential and saturable nonlinearity.
References
C. O. Alves, N. V. Thin: On existence of multiple normalized solutions to a class of elliptic problems in whole $R^N$ via Lusternik-Schnirelmann category, SIAM J. Math. Anal., 55 (2023), 1264–1283.
C. O. Alves, C. Ji: Multiple normalized solutions to a logarithmic Schrödinger equation via Lusternik–Schnirelmann category, J. Geom. Anal., 34 (2024), Article ID: 198.
L. Appolloni, S. Secchi: Normalized solutions for the fractional NLS with mass supercritical nonlinearity, J. Differential Equations, 286 (2021), 248–283.
T. Bartsch, Z. Q. Wang: Existence and multiplicity results for some superlinear elliptic problems on $R^n$, Comm. Partial Differential Equations, 20 (1995), 1725–1741.
H. Berestycki, P. L. Lions: Nonlinear scalar field equations I: existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313–345.
H. Berestycki, P. L. Lions: The concentration-compactness principle in the calculus of variations. The locally compact case, part 1, Ann. Inst. Henri. Poincaré Anal. Non Linéaire, 11 (1984), 109–145.
H. Brézis, E. Lieb: A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc., 88 (3) (1983), 486–490.
N. K. Efremidis, J. Hudock, D. N. Christodoulides, J.W. Fleischer, O. Cohen and M. Segev: Two-dimensional optical lattice solitons, Phys Rev Lett., 91 (2003), 213–906.
R. L. Frank, E. Lenzmann and L. Silvestre: Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671–1726.
Q. Y. Guan, Z. M. Ma: Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385–424.
N. Laskin: Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), Article ID: 56108.
S. Liang, P. Pucci and B. Zhang: Multiple solutions for critical Choquard-Kirchhoff type equations, Adv. Nonlinear Anal., 10 (2021), 400–419.
Q. Li, W. Zou: The existence and multiplicity of the normalized solutions for fractional Schrödinger equations involving Sobolev critical exponent in the $L^2$-subcritical and $L^2$-supercritical cases, Adv. Nonlinear Anal., 11 (2022), 1531–1551.
T. C. Lin, M. R. Beli´c, M. S. Petrovi´c and G. Chen: Ground states of nonlinear Schrödinger systems with saturable nonlinearity in $R^2$ for two counterpropagating beams, J. Math. Phys., 55 (2014), Article ID: 011505.
T. C. Lin, X. Wang and Z. Q. Wang: Orbital stability and energy estimate of ground states of saturable nonlinear Schrödinger equations with intensity functions in $R^2$, J. Diff. Equ., 263 (2017), 2750–2786.
T. C. Lin, T. F. Wu: Multiple positive solutions of saturable nonlinear Schrödinger equations with intensity functions, Discrete Contin. Dyn. Syst., 70 (2020), 2165–2187.
H. Luo, Z. Zhang: Normalized solutions to the fractional Schrödinger equations with combined nonlinearities, Calc. Var. Partial Differ. Equ., 59 (2020), Article ID: 143.
Y-C Lv, G-D Li: Multiplicity of normalized solutions to a fractional logarithmic Schrödinger equation,Fractal fract., 8 (7) (2024), Article ID: 391.
E. D. Nezza, G. Palatucci and E. Valdinoci: Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573.
L. Silvestre: Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67–112.
Y. Sire, E. Valdinoci: Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842–1864.
Y. Zhang, J. Sun: Normalized solutions of quasilinear Schrödinger equations with saturable nonlinearity, Appl. Math. Lett., 138 (2023), Article ID: 108531.
M. Zhen, B. Zhang: Normalized ground states for the critical fractional NLS equation with a perturbation, Rev. Mat. Complut., 35 (2022), 89–132.
J. Zuo, T. An and A. Fiscella: A critical Kirchhoff-type problem driven by a $p(·)$-fractional Laplace operator with variable $s(·)$-order, Math. Methods Appl. Sci., 44 (2021), 1071–1085.
J. B. Zuo, C. G. Liu and C. Vetro: Normalized solutions to the fractional Schrödinger equation with potential, Mediterr. J. Math., 216 (2023), 1660–5446.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Wen Liao, Qiongfen Zhang
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
