A NONLINEAR SCHR&#214;DINGER EQUATION WITH COULOMB POTENTIAL

Changxing MIAO; Junyong ZHANG; Jiqiang ZHENG

doi:10.1007/s10473-022-0606-x

Acta mathematica scientia, Series B >

2022 , Vol. 42 >Issue 6: 2230 - 2256

DOI: https://doi.org/10.1007/s10473-022-0606-x

Articles

A NONLINEAR SCHRÖDINGER EQUATION WITH COULOMB POTENTIAL

Changxing MIAO ,
Junyong ZHANG ,
Jiqiang ZHENG

Expand

1. Institute of Applied Physics and Computational Mathematics, Beijing 100088, China;
2. Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China;
3. Department of Mathematics, Cardiff University, UK

Received date: 2022-07-01

Online published: 2022-12-16

Supported by

The authors were supported by NSFC (12126409, 12026407, 11831004) and the J. Zheng was also supported by Beijing Natural Science Foundation (1222019).

Fold

Abstract

In this paper, we study the Cauchy problem for the nonlinear Schrödinger equations with Coulomb potential i$?_tu+\Delta u+\frac{K}{|x|}u=\lambda|u|^{p-1}u$ with 1<p≤5 on $\mathbb{R}^3$. Our results reveal the influence of the long range potential $K|x|^{-1}$ on the existence and scattering theories for nonlinear Schrödinger equations. In particular, we prove the global existence when the Coulomb potential is attractive, i.e., when $K>0$, and the scattering theory when the Coulomb potential is repulsive, i.e., when $K\leq0$. The argument is based on the newly-established interaction Morawetz-type inequalities and the equivalence of Sobolev norms for the Laplacian operator with the Coulomb potential.

Key words： nonlinear Schrödinger equations; long range potential; global well-posedness; blow-up; scattering

Cite this article

Changxing MIAO , Junyong ZHANG , Jiqiang ZHENG . A NONLINEAR SCHRÖDINGER EQUATION WITH COULOMB POTENTIAL[J]. Acta mathematica scientia, Series B, 2022 , 42(6) : 2230 -2256 . DOI: 10.1007/s10473-022-0606-x

References

[1] Berestycki H, Lions P L. Nonlinear scalar field equations. I Existence of a ground state. Arch Rational Mech Anal, 1983, 82(4): 313-345
[2] Benguria R, Brezis H, Lieb E H. The Thomas-Fermi-von Weizsacker theory of atoms and molecules. Commun Math Phys, 1981, 79: 167-180
[3] Benguria R, Jeanneret L. Existence and uniqueness of positive solutions of semilinear elliptic equations with Coulomb potentials on R³. Commun Math Phys, 1986, 104: 291-306
[4] J. Bourgain, Global well-posedness of defocusing 3D critical NLS in the radial case. J Amer Math Soc, 1999, 12: 145-171
[5] Cazenave T. Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, Vol 10. New York: New York University Courant Institute of Mathematical Sciences, 2003
[6] Chadam J M, Glassey R T. Global existence of solutions to the Cauchy problem for time-dependent Hartree equations. J Math Phys, 1975, 16: 1122-1130
[7] Colliander J, Keel M, Staffilani G, Takaoka H, Tao T. Global existence and scattering for rough solutions of a nonlinear Schrödinger equations on R³. Comm Pure Appl Math, 2004, 57: 987-1014
[8] Colliander J, Keel M, Staffilani G, Takaoka H, Tao T. Global well-posedness and scattering for the energycirtical nonlinear Schrödinger equation in R³. Annals of Math, 2008, 167: 767-865
[9] Christ M, Weinstein M. Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation. J Funct Anal, 1991, 100: 87-109
[10] Dias J, Figueira M. Conservation laws and time decay for the solutions of some nonlinear SchrödingerHartree equations and systems. J Math Anal Appl, 1981, 84: 486-508
[11] Dodson B. Global well-posedness and scattering for the focusing, energy-critical nonlinear Schrödinger problem in dimension d = 4 for initial data below a ground state threshold. Ann Scient Éc Norm Sup, 2019, 52: 139-180
[12] Dodson B, Murphy J. A new proof of scattering belw the ground state for the 3D radial focusing NLS. Proc Amer Math Soc, 2017, 145(11): 4859-4867
[13] Dodson B, Murphy J. A new proof of scattering below the ground state for the non-radial focusing NLS. Math Res Lett, 2018, 25(6): 1805-1825
[14] Duyckaerts T, Holmer J, Roudenko S. Scattering for the non-radial 3D cubic nonlinear Schrödinger equation. Math Res Lett, 2008, 15(6): 1233-1250
[15] Gidas B, Ni W M, Nirenberg L. Symmetry and related properties via the maximum principle. Comm Math Phys, 1979, 68: 209-243
[16] Ginibre J, Velo G. On the class of nonlinear Schrödinger equation I & II. J Funct Anal, 1979, 32: 1-72
[17] Ginibre J, Velo G. Scattering theory in the energy space for a class of nonlinear Schrödinger equations. J Math Pure Appl, 1985, 64: 363-401
[18] Hayashi N, Ozawa T. Time decay of solutions to the Cauchy problem for time-dependent SchrödingerHartree equations. Commun Math Phys, 1987, 110: 467-478
[19] Holmer J, Roudenko S. A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation. Comm Math Phys, 2008, 282(2): 435-467
[20] Hong Y. Scattering for a nonlinear Schrödinger equation with a potential. Commun Pure Appl Anal, 2016, 15(5): 1571-1601
[21] Isozaki H, Kitada H. Modified wave operators with time independent modifiers. J Fac Sci Univ Tokyo, 1985, 32: 77-104
[22] Keel M, Tao T. Endpoint Strichartz estimates. Amer J Math, 1998, 120: 955-980
[23] Kenig C, Merle F. Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case. Invent Math, 2006, 166(3): 645-675
[24] Killip R, Miao C, Visan M, Zhang J, Zheng J. Sobolev spaces adapted to the Schrödinger operator with inverse-square potential. Math Z, 2018, 288(3/4): 1273-1298
[25] Killip R, Murphy J, Visan M, Zheng J. The focusing cubic NLS with inverse square potential in three space dimensions. Diff Inte Equ, 2017, 30: 161-206
[26] Killip R, Visan M. The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. Amer J Math, 2010, 132: 361-424
[27] Kurata K. An estimate on the heat kernel of magnetic Schrödinger operators and uniformly elliptic operators with non-negative potentials. J Lond Math Soc, 2000, 62: 885-903
[28] Kwong M. Uniqueness of positive solutions of ?u-u+u^p=0 in Rⁿ. Arch Rational Mech Anal, 1989, 105(3): 243-266
[29] Lenzmann E, Lewin M. Dynamical ionization bounds for atoms. Analysis and PDE, 2013, 6: 1183-1211
[30] Lieb E H. Thomas-Fermi and related theories of atoms and molecules. Rev Mod Phys, 1981, 53: 603-641
[31] Lu J, Miao C, Murphy J. Scattering in H¹ for the intercritical NLS with an inverse-square potential. J Differential Equations, 2018, 264: 3174-3211
[32] Messiah A. Quantum Mechanics. Amsterdam: North Holland, 1961
[33] Mizutani H. Strichartz estimates for Schrödinger equations with slowly decaying potentials. J Funct Anal, 2022, 279(12): 1087891808.06987v1.
[34] O’Neil R. Convolution operators and L(p, q) spaces. Duke Math J, 1963, 30: 129-142
[35] Planchon F. On the Cauchy problem in Besov spaces for a non-linear Schrödinger equation. Communications in Contemporary Mathematics, 2000, 2(2): 243-254
[36] Reed M, Simon B. Methods of Mathematical Physics. Vols 1, 2. New York: Academic, 1975; Vols 3, 4. 1978
[37] Ryckman E, Visan M. Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R¹⁺⁴. Amer J Math, 2007, 129: 1-60
[38] Series G. Spectrum of Atomic Hydrogen. Oxford: Oxford University Press, 1957
[39] Sikora A, Wright J. Imaginary powers of Laplace operators. Proc Amer Math Soc, 2001, 129: 1745-1754
[40] Shen Z. L^p estimates for Schrödinger operators with certain potentials. Ann Inst Fourier (Greno-ble), 1995, 45: 513-546
[41] Tao T. Nonlinear Dispersive Equations, Local and Global Analysis. CBMS Reg Conf Ser Math, Vol 106. Providence, RI: Amer Math Soc, 2006; Washington, DC: published for the Conference Board of the Mathematical Science
[42] Taylor M. Partial Differential Equations, Vol II. Springer, 1996
[43] Visan M. The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions. Duke Math J, 2007, 138: 281-374
[44] Zhang, Zheng J. Scattering theory for nonlinear Schrödinger with inverse-square potential. J Funct Anal, 2014, 267: 2907-2932
[45] Zhang X. On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations. J Differential Equations, 2006, 230: 422-445

Options

Abstract

Outlines

模态框（Modal）标题

Abstract

Cite this article

References