On nonlinear Schrödinger equations with attractive inverse-power potentials
Keywords
Nonlinear Schrödinger equation, inverse-power potential, standing waves, stability, global well-posedness, blow-upAbstract
We study the Cauchy problem for nonlinear Schrödinger equations with attractive inverse-power potentials. By using variational arguments, we first determine a sharp threshold of global well-posedness and blow-up for the equation in the mass-supercritical case. We next study the existence and orbital stability of standing waves for the problem in the mass-subcritical and mass-critical cases. In the mass-critical case, we give a detailed description of the blow-up behavior of standing waves when the mass tends to a critical value.References
A.H. Ardila and V.D. Dinh, Some qualitative studies of the focusing inhomogeneous Gross–Pitaevskiı̆ equation, Z. Angew. Math. Phys. 71 (2020), 79.
J. Bellazzini, N. Boussaid, L. Jeanjean and N. Visciglia, Existence and stability of standing waves for supercritical NLS with a partial confinement, Comm. Math. Phys. 353 (2017), 229–339.
R. Benguria and L. Jeanneret, Existence and uniqueness of positive solutions of semilinear elliptic equations with coulomb potentials on R3 , Commun. Math. Phys. 104 (1986), 291–306.
A. Bensouilah, V.D. Dinh and S. Zhu, On stability and instability of standing waves for the nonlinear Schrödinger equation with an inverse-square potential, J. Math. Phys. 59 (2018), 101505.
H. Berestycki and P.L. Lions, Nonlinear scalar field equations, I: Existence of ground state, Arch. Ration. Mech. Anal. 82 (1983), 313–345.
N. Burq, F. Planchon, J. Stalker and A.S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with inverse-square pontential, J. Funct. Anal. 203 (2003), 519–549.
H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.
J. M. Chadam and R.T. Glassey, Global existence of solutions to the Cauchy problem for time-dependent Hartree equations, J. Math. Phys. 16 (1975), 1122–1130.
E. Csobo and F. Genoud, Minimal mass blow-up solutions for the L2 critical NLS with inverse-square potential, Nonlinear Anal. 168 (2018), 110–129.
C.R. de Oliveira, Intermediate Spectral Theory and Quantum Dynamics, Progress in Mathematical Physics, vol. 54, Birkhäuser, Berlin, 2009.
V.D. Dinh, Global existence and blowup for a class of the focusing nonlinear Schrödinger equation with inverse-square potential, J. Math. Anal. Appl. 468 (2018), 270–303.
V.D. Dinh, On instability of radial standing waves for the nonlinear Schrödinger equation with inverse-square potential, Complex Var. Elliptic Equ. 2020 (in press).
V.D. Dinh, On nonlinear Schrödinger equations with repulsive inverse-power potentials, Acta Appl. Math. 171 (2021), article no. 14.
N. Fukaya and M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with attractive inverse power potential, Osaka J. Math. 56 (2019), 713–726.
R. Fukuizumi and M. Ohta, Instability of standing waves for nonlinear Schrödinger equations with potentials, Differential Integral Equations 16 (2003), 691–706.
R.T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys. 18 (1977), 1794–1797.
D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, third edition, Springer–Verlag, Berlin, 2001.
Y. Guo and R. Seiringer, On the mass concentration for Bose–Einstein condensates with attractive interactions, Lett. Math. Phys. 104 (2014), 141–156.
Y.J. Guo, X.Y. Zeng and H.S. Zou, Energy estimates and symmetry breaking in attractive Bose–Einstein condensates with ring-shaped potentials, Ann. Inst. Henri Poincaré Nonlinear Anal. 33 (2016), 809–828.
Q. Guo, H. Wang and X. Yao, Dynamics of the focusing 3D cubic NLS with slowly decaying potential, 2018, https://arxiv.org/abs/1811.07578, preprint.
H. Hajaiej, Cases of equality and strict inequality in the extended Hardy–Littlewood inequalities, Proc. Roy. Soc. Edinburgh Sect. A 135 (2005), 643–661.
N. Hayashi and T. Ozawa, Time decay of solutions to the Cauchy problem for timedependent Schrödinger–Hartree equations, Commun. Math. Phys. 110 (1987), 467–478.
R. Killip, C. Miao, M. Visan, J. Zhang and J. Zheng, The energy-critical NLS with inverse-square potential, Discrete Contin. Dyn. Syst. 37 (2017), 3831–3866.
R. Killip, J. Murphy, M. Visan and J. Zheng, The focusing cubic NLS with inversesquare potential in three space dimension, Differential Integral Equations 30 (2017), 161–206.
R. Killip, C. Miao, M. Visan, J. Zhang and J. Zheng, Sobolev spaces adapted to the Schrödinger operator with inverse-square potential, Math. Z. 288 (2018), 1273–1298.
E.H. Lieb and M. Loss, Analysis, second edition, Graduate Studies in Mathematics, vol. 14, AMS, Providence, 2001.
X. Li and J. Zhao, Orbital stability of standing waves for Schrödinger type equations with slowly decaying linear potential, Comput. Math. Appl. 79 (2020), 303–316.
P.L. Lions, Some remarks on Hartree equation, Nonlinear Anal. 5 (1981), 1245–1256.
P.L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part 1, Ann. Inst. Henri Poincaré 1 (1984), 109–145.
J. Lu, C. Miao and J. Murphy, Scattering in H 1 for the intercritical NLS with an inverse-square potential, J. Differential Equations 264 (2018), 3174–3211.
C. Miao, J. Zhang and J. Zheng, Nonlinear Schrödinger equation with coulomb potential, preprint https://arxiv.org/abs/1809.06685, 2018.
H. Mizutani, Strichartz estimates for Schrödinger equations with slowly decaying potentials, J. Func. Anal. 279 (2020), no. 12, 108789.
N. Okazawa, T. Suzuki and T. Yokota, Energy methods for abstract nonlinear Schrödinger equations, Evol. Equ. Control Theory 1, 337–354.
T.V. Phan, Blow-up profile of Bose-Einstein condensate with singular potentials, J. Math. Phys. 58 (2017), 072301.
N. Shioji and K. Watanabe, A generalized Pohozaev identity and uniqueness of positive radial solutions of ∆u + g(r)u + h(r)up = 0, J. Differential Equations 255 (2013), 4448–4475.
Q. Wang and D. Zhao, Existence and mass concentration of 2D attractive Bose–Einstein condenstates with periodic potentials, J. Differential Equations 262 (2017), 2684–2704.
M.I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys. 87 (1983), 567–576.
X. Zhang, On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations, J. Differential Equations 230 (2006), 422-445.
J. Zhang and J. Zheng, Scattering theory for nonlinear Schrödinger equation with inverse-square potential, J. Funct. Anal. 267 (2014), 2907–2932.
J. Zheng, Focusing NLS with inverse square potential, J. Math. Phys. 59 (2018), 111502.
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