Blow-up criteria and instability of standing waves for the fractional Schrodinger Poisson equation
DOI:
https://doi.org/10.58997/ejde.2023.24Keywords:
Schrodinger-Poisson equation, blow-up criteria, strong instability, standing waves, well-posednessAbstract
In this article, we consider blow-up criteria and instability of standing waves for the fractional Schrodinger-Poisson equation. By using the localized virial estimates, we establish the blow-up criteria for non-radial solutions in both mass-critical and mass-supercritical cases. Based on these blow-up criteria and three variational characterizations of the ground state, we prove that the standing waves are strongly unstable. These obtained results extend the corresponding ones presented in the literature.
Author Biographies
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Yichun Mo
Department of Mathematics
Northwest Normal University
Lanzhou 730070, China
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Min Zhu
Department of Mathematics
Nanjing Forestry University
Nanjing, Jiangsu 210037, China
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Binhua Feng
Department of Mathematics
Northwest Normal University
Lanzhou 730070, China
References
J. Bellazzini, G. Siciliano; Stable standing waves for a class of nonlinear Schrödinger-Poisson equations, Z. Angew. Math. Phys. (2011), 267-280. https://doi.org/10.1007/s00033-010-0092-1
J. Bellazzini, L. Jeanjean, T. Luo; Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations, Proc. Lond. Math. Soc., 107 (2013), 303- 339. https://doi.org/10.1112/plms/pds072
T. Boulenger, D. Himmelsbach, E. Lenzmann; Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603. https://doi.org/10.1016/j.jfa.2016.08.011
H. Brézis, E. H. Lieb; A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. https://doi.org/10.1090/S0002-9939-1983-0699419-3
T. Cazenave; Semilinear Schrodinger equations, Courant Lecture Notes in Mathematics vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
Y. Cho, G. Hwang, S. Kwon, S. Lee; On finite time blow-up for the mass-critical Hartree equations, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 467-479. https://doi.org/10.1017/S030821051300142X
V. D. Dinh; Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces, Int. J. Appl. Math., 31 (2018), 483-525. https://doi.org/10.12732/ijam.v31i4.1
V. D. Dinh, B. Feng; On fractional nonlinear Schrödinger equation with combined power-type nonlinearities, Discrete Contin. Dyn. Syst., 29 (2019), 4565-4612. https://doi.org/10.3934/dcds.2019188
D. Du, Y. Wu, K. Zhang; On blow-up criterion for the nonlinear Schrödinger equation, Discrete Contin. Dyn. Syst., 36 (2016), 3639-3650. https://doi.org/10.3934/dcds.2016.36.3639
B. Feng; On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 1785-1804 https://doi.org/10.3934/cpaa.2018085
B. Feng; On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities, J. Evol. Equ., 18 (2018), 203-220. https://doi.org/10.1007/s00028-017-0397-z
B. Feng, R. Chen, J. Liu; Blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation, Adv. Nonlinear Anal., 10 (2021), 311-330. https://doi.org/10.1515/anona-2020-0127
B. Feng, R. Chen, Q. Wang; Instability of standing waves for the nonlinear Schrödinger- Poisson equation in the L2-critical case, J. Dynam. Differential Equations, 32 (2020), 1357- 1370. https://doi.org/10.1007/s10884-019-09779-6
B. Feng, Q. Wang; Strong instability of standing waves for the nonlinear Schrödinger equation in trapped dipolar quantum gases, J. Dynam. Differential Equations, 33 (2021), 1989-2008. https://doi.org/10.1007/s10884-020-09881-0
J. Fröhlich, G. Jonsson, E. Lenzmann; Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30. https://doi.org/10.1007/s00220-007-0272-9
R. Fukuizumi; Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential, Discrete Contin. Dyn. Syst., 7 (2001), 525-544. https://doi.org/10.3934/dcds.2001.7.525
R. Fukuizumi, M. Ohta; Instability of standing waves for nonlinear Schrödinger equations with potentials, Differential Integral Equations, 16 (2003), 691-706. https://doi.org/10.57262/die/1356060607
N. Fukaya, M. Ohta; Strong instability of standing waves for nonlinear Schrödinger equations with attractive inverse power potential, Osaka J. Math., 56 (2019), 713-726.
N. Fukaya, M. Ohta; Strong instability of standing waves with negative energy for double power nonlinear Schrödinger equations, SUT J. Math., 54 (2018), 131-143. https://doi.org/10.55937/sut/1549709992
Y. Hong, Y. Sire; On fractional Schrödinger equations in Sobolev spaces, Comm. Pure Appl. Anal., 14 (2015), 2265-2282. https://doi.org/10.3934/cpaa.2015.14.2265
L. Jeanjean, T. Luo; Sharp nonexistence results of prescribed L2-norm solutions for some class of Schrödinger-Poisson and quasi-linear equations, Z. Angew. Math. Phys., 64 (2013), 937-954. https://doi.org/10.1007/s00033-012-0272-2
K. Kirkpatrick, E. Lenzmann, G. Staffilani; On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317 (2013), 563-591. https://doi.org/10.1007/s00220-012-1621-x
N. Laskin; Fractional Quantum Mechanics and Lèvy Path Integrals, Phys. Lett. A, 268 (2000), 298-304. https://doi.org/10.1016/S0375-9601(00)00201-2
N. Laskin; Fractional Schrödinger equations, Physics Review E, 66 (2002), 056108. https://doi.org/10.1103/PhysRevE.66.056108
H. Kikuchi; Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7 (2007), 403-437. https://doi.org/10.1515/ans-2007-0305
S. Le Coz; A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations, Adv. Nonlinear Stud., 8 (2008), 455-463. https://doi.org/10.1515/ans-2008-0302
X. Lin, S. Zheng; Mixed local and nonlocal Schrodinger-Poisson type system involving variable exponents, Electron. J. Differential Equations, 2022 (2022), no. 81, 1-17.
S. Longhi; Fractional Schrödinger equation in optics, Opt. Lett., 40 (2015), 1117-1120. https://doi.org/10.1364/OL.40.001117
M. Ohta; Strong instability of standing waves for nonlinear Schrödinger equations with harmonic potential, Funkcial. Ekvac., 61 (2018), 135-143. https://doi.org/10.1619/fesi.61.135
M. Ohta; Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement, Comm. Pure Appl. Anal., 17 (2018), 1671-1680. https://doi.org/10.3934/cpaa.2018080
M. Ohta; Stability of standing waves for the generalized Davey-Stewartson system, J. Dynam. Differential Equations, 6 (1994), 325-334. https://doi.org/10.1007/BF02218533
M. Ohta; Stability and instability of standing waves for the generalized Davey-Stewartson system, Differential Integral Equations, 8 (1995), 1775-1788. https://doi.org/10.57262/die/1368397756
C. Peng, Q. Shi; Stability of standing wave for the fractional nonlinear Schrödinger equation, J. Math. Phys., 59 (2018), 011508. https://doi.org/10.1063/1.5021689
S. Qu, X. He; Multiplicity of high energy solutions for fractional Schrodinger-Poisson systems with critical frequency, Electron. J. Differential Equations, 2022 (2022), no. 47, 1-21.
T. Saanouni; Strong instability of standing waves for the fractional Schrödinger equation, J. Math. Phys., 59 (2018), 081509. https://doi.org/10.1063/1.5043473
T. Saanouni; Remarks on the inhomogeneous fractional nonlinear Schrödinger equation, J. Math. Phys., 57 (2016), 081503. https://doi.org/10.1063/1.4960045
Q. Shi, C. Peng, Q. Wang; Blowup results for the fractional Schrödinger equation without gauge invariance, Discrete Contin. Dyn. Syst. Ser. B, 27 (2022), 6009-6022. https://doi.org/10.3934/dcdsb.2021304
J. Sun, T. F. Wu, Z. Feng; Multiplicity of positive solutions for a nonlinear Schrödinger- Poisson system, J. Differential Equations, 260 (2016), 586-627. https://doi.org/10.1016/j.jde.2015.09.002
T. Tao, M. Visan, X. Zhang; The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343. https://doi.org/10.1080/03605300701588805
K. Teng; Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2016), 3061-3106. https://doi.org/10.1016/j.jde.2016.05.022
J. Zhang; Sharp threshold for blowup and global existence in nonlinear Schrödinger equations under a harmonic potential, Comm. Partial Differential Equations, 30 (2007), 1429-1443. https://doi.org/10.1080/03605300500299539
S. Zhu; On the Davey-Stewartson system with competing nonlinearities, J. Math. Phys., 57 (2016), 031501. https://doi.org/10.1063/1.4942633
S. Zhu; On the blow-up solutions for the nonlinear fractional Schrödinger equation, J. Differential Equations, 261 (2016), 1506-1531. https://doi.org/10.1016/j.jde.2016.04.007
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