The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data

Van Duong Dinh; Sahbi Keraani

doi:10.3934/dcdss.2020407

Previous Article
Numerical solutions for a Timoshenko-type system with thermoelasticity with second sound
DCDS-S Home
This Issue
Next Article
Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point

doi: 10.3934/dcdss.2020407

The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data

Van Duong Dinh ^1,2, and Sahbi Keraani ^1,,

1.	Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France
2.	Department of Mathematics, HCMC University of Pedagogy, 280 An Duong Vuong, Ho Chi Minh, Vietnam

^* Corresponding author: Sahbi Keraani

Received January 2020 Revised June 2020 Published July 2020

Based on recent works of Dodson-Murphy [12] and Arora-Dodson-Murphy [3], we give a unified approach for the energy scattering with radially symmetric initial data for nonlinear Schrödinger equations and nonlinear Choquard equations in any dimensions $ N\geq 2 $. We also discuss its applications for other Schrödinger-type equations.

Keywords: Nonlinear Schrödinger equation, Nonlinear Choquard equation, Ground state, Scattering, Radial Sobolev embedding.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35P25.

Citation: Van Duong Dinh, Sahbi Keraani. The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020407

References:

[1]	T. Akahori and H. Nawa, Blowup and scattering problems for the nonlinear Schrödinger equations, Kyoto J. Math., 53 (2013), 629-672. doi: 10.1215/21562261-2265914. Google Scholar
[2]	A. K. Arora, Scattering of radial data in the focusing NLS and generalized Hartree equations, Discrete Cont. Dyn. Syst., 39 (2019), 6643-6668. doi: 10.3934/dcds.2019289. Google Scholar
[3]	A. K. Arora, B. Dodson and J. Murphy, Scattering below the ground state for the 2D radial nonlinear Schrödinger equation, Proc. Amer. Math. Soc., 148 (2020), 1653-1663. doi: 10.1090/proc/14824. Google Scholar
[4]	A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, preprint, arXiv: 1904.05339. Google Scholar
[5]	L. Campos, Scattering of radial solutions to the inhomogeneous nonlinear Schrödinger equation, preprint, arXiv: 1905.02663. Google Scholar
[6]	T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, AMS, 2003. doi: 10.1090/cln/010. Google Scholar
[7]	T. Cazenave, D. Fang and J. Xie, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062. doi: 10.1007/s11425-011-4283-9. Google Scholar
[8]	T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Commun. Math. Phys., 147 (1992), 75-100. doi: 10.1007/BF02099529. Google Scholar
[9]	C. V. Coffman, Uniqueness of the ground state solution for $\Delta u - u + u^3 = 0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95. doi: 10.1007/BF00250684. Google Scholar
[10]	V. D. Dinh, Energy scattering for a class of inhomogeneous nonlinear Schrödinger equation in two dimensions, preprint, arXiv: 1908.02987. Google Scholar
[11]	V. D. Dinh, Dynamics of radial solutions for the focusing fourth-order Schrödinger equations, preprint, arXiv: 2001.03022. Google Scholar
[12]	B. Dodson and J. Murphy, A new proof of scattering below the ground state for the 3D radial focusing NLS, Proc. Amer. Math. Soc., 145 (2017), 4859-4867. doi: 10.1090/proc/13678. Google Scholar
[13]	B. Dodson and J. Murphy, A new proof of scattering below the ground state for the non-radial focusing NLS, Math. Res. Lett., 25 (2018), 1805-1825. doi: 10.4310/MRL.2018.v25.n6.a5. Google Scholar
[14]	T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250. doi: 10.4310/MRL.2008.v15.n6.a13. Google Scholar
[15]	L. G. Farah and C. Guzman, Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schrödinger equation, J. Differential Equations, 262 (2017), 4175-4231. doi: 10.1016/j.jde.2017.01.013. Google Scholar
[16]	L. G. Farah and C. Guzman, Scattering for the radial focusing INLS equation in higher dimensions, Bull. Braz. Math. Soc., 51 (2020), 449-512. doi: 10.1007/s00574-019-00160-1. Google Scholar
[17]	D. Foschi, Inhomogeneous strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24. doi: 10.1142/S0219891605000361. Google Scholar
[18]	C. Guevara, Global behavior of finite energy solutions to the d-dimensional focusing nonlinear Schrödinger equation, Appl. Math. Res. Express., 2014 (2014), 177-243. Google Scholar
[19]	Q. Guo, Scattering for the focusing $L^2$-supercritical and $\dot{H}^2$-subcritical biharmonic NLS equations, Comm. Partial Differential Equations, 41 (2016), 185-207. doi: 10.1080/03605302.2015.1116556. Google Scholar
[20]	Q. Guo and S. Zhu, Sharp criteria of scattering for the fractional NLS, preprint, arXiv: 1706.02549. Google Scholar
[21]	M. Hamano and M. Ikeda, Global dynamics below the ground state for the focusing Schrödinger equation with a potential, J. Evol. Equ., 2019 (in press). Google Scholar
[22]	J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467. doi: 10.1007/s00220-008-0529-y. Google Scholar
[23]	Y. Hong, Scattering for a nonlinear Schrödinger equation with a potential, Commun. Pure Appl. Anal., 15 (2016), 1571-1601. doi: 10.3934/cpaa.2016003. Google Scholar
[24]	M. Keel and T. Tao, Endpoint strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039. Google Scholar
[25]	C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical, focusing, nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4. Google Scholar
[26]	J. Krieger, E. Lenzmann and P. Raphaël, On stability of pseudo-conformal blowup for $L^2$-critical Hartree NLS, Ann. Henri Poincaré, 10 (2009), 1159-1205. doi: 10.1007/s00023-009-0010-2. Google Scholar
[27]	M. K. Kwong, Uniqueness of positive solution of $\Delta u - u + u^p = 0$ in $ \mathbb R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. Google Scholar
[28]	E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105. doi: 10.1002/sapm197757293. Google Scholar
[29]	C. Miao, G. Xu and L. Zhao, The cauchy problem of the hartree equation, J. Partial Diff. Eqs., 21 (2008), 22-44. Google Scholar
[30]	V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184. doi: 10.1016/j.jfa.2013.04.007. Google Scholar
[31]	T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330. doi: 10.1016/0022-0396(91)90052-B. Google Scholar
[32]	C. Sun, H. Wang, X. Yao and J. Zheng, Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 38 (2018), 2207-2228. doi: 10.3934/dcds.2018091. Google Scholar
[33]	E. Stein and T. S. Murphy, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton University Press, Princeton, 1993. Google Scholar
[34]	J. Stubbe, Global solutions and stable ground states of nonlinear Schrödinger equations, Phys. D, 48 (1991), 259-272. doi: 10.1016/0167-2789(91)90087-P. Google Scholar
[35]	W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517. Google Scholar
[36]	T. Tao, On the asymptotic behavior of large radial data for a focusing nonlinear Schrödinger equation, Dyn. Partial Differ. Equ., 1 (2004), 1-48. doi: 10.4310/DPDE.2004.v1.n1.a1. Google Scholar
[37]	T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343. doi: 10.1080/03605300701588805. Google Scholar
[38]	M. C. Vilela, Inhomogeneous strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136. doi: 10.1090/S0002-9947-06-04099-2. Google Scholar
[39]	M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. Google Scholar
[40]	C. L. Xiang, Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions, Calc. Var. Partial Differential Equations, 55 (2016), Art. 134, 25 pp. doi: 10.1007/s00526-016-1068-6. Google Scholar
[41]	C. Xu and T. Zhao, A remark on the scattering theory for the 2D radial focusing INLS, preprint, arXiv: 1908.00743. Google Scholar

show all references

References:

[1]	T. Akahori and H. Nawa, Blowup and scattering problems for the nonlinear Schrödinger equations, Kyoto J. Math., 53 (2013), 629-672. doi: 10.1215/21562261-2265914. Google Scholar
[2]	A. K. Arora, Scattering of radial data in the focusing NLS and generalized Hartree equations, Discrete Cont. Dyn. Syst., 39 (2019), 6643-6668. doi: 10.3934/dcds.2019289. Google Scholar
[3]	A. K. Arora, B. Dodson and J. Murphy, Scattering below the ground state for the 2D radial nonlinear Schrödinger equation, Proc. Amer. Math. Soc., 148 (2020), 1653-1663. doi: 10.1090/proc/14824. Google Scholar
[4]	A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, preprint, arXiv: 1904.05339. Google Scholar
[5]	L. Campos, Scattering of radial solutions to the inhomogeneous nonlinear Schrödinger equation, preprint, arXiv: 1905.02663. Google Scholar
[6]	T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, New York University, AMS, 2003. doi: 10.1090/cln/010. Google Scholar
[7]	T. Cazenave, D. Fang and J. Xie, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062. doi: 10.1007/s11425-011-4283-9. Google Scholar
[8]	T. Cazenave and F. B. Weissler, Rapidly decaying solutions of the nonlinear Schrödinger equation, Commun. Math. Phys., 147 (1992), 75-100. doi: 10.1007/BF02099529. Google Scholar
[9]	C. V. Coffman, Uniqueness of the ground state solution for $\Delta u - u + u^3 = 0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95. doi: 10.1007/BF00250684. Google Scholar
[10]	V. D. Dinh, Energy scattering for a class of inhomogeneous nonlinear Schrödinger equation in two dimensions, preprint, arXiv: 1908.02987. Google Scholar
[11]	V. D. Dinh, Dynamics of radial solutions for the focusing fourth-order Schrödinger equations, preprint, arXiv: 2001.03022. Google Scholar
[12]	B. Dodson and J. Murphy, A new proof of scattering below the ground state for the 3D radial focusing NLS, Proc. Amer. Math. Soc., 145 (2017), 4859-4867. doi: 10.1090/proc/13678. Google Scholar
[13]	B. Dodson and J. Murphy, A new proof of scattering below the ground state for the non-radial focusing NLS, Math. Res. Lett., 25 (2018), 1805-1825. doi: 10.4310/MRL.2018.v25.n6.a5. Google Scholar
[14]	T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250. doi: 10.4310/MRL.2008.v15.n6.a13. Google Scholar
[15]	L. G. Farah and C. Guzman, Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schrödinger equation, J. Differential Equations, 262 (2017), 4175-4231. doi: 10.1016/j.jde.2017.01.013. Google Scholar
[16]	L. G. Farah and C. Guzman, Scattering for the radial focusing INLS equation in higher dimensions, Bull. Braz. Math. Soc., 51 (2020), 449-512. doi: 10.1007/s00574-019-00160-1. Google Scholar
[17]	D. Foschi, Inhomogeneous strichartz estimates, J. Hyperbolic Differ. Equ., 2 (2005), 1-24. doi: 10.1142/S0219891605000361. Google Scholar
[18]	C. Guevara, Global behavior of finite energy solutions to the d-dimensional focusing nonlinear Schrödinger equation, Appl. Math. Res. Express., 2014 (2014), 177-243. Google Scholar
[19]	Q. Guo, Scattering for the focusing $L^2$-supercritical and $\dot{H}^2$-subcritical biharmonic NLS equations, Comm. Partial Differential Equations, 41 (2016), 185-207. doi: 10.1080/03605302.2015.1116556. Google Scholar
[20]	Q. Guo and S. Zhu, Sharp criteria of scattering for the fractional NLS, preprint, arXiv: 1706.02549. Google Scholar
[21]	M. Hamano and M. Ikeda, Global dynamics below the ground state for the focusing Schrödinger equation with a potential, J. Evol. Equ., 2019 (in press). Google Scholar
[22]	J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467. doi: 10.1007/s00220-008-0529-y. Google Scholar
[23]	Y. Hong, Scattering for a nonlinear Schrödinger equation with a potential, Commun. Pure Appl. Anal., 15 (2016), 1571-1601. doi: 10.3934/cpaa.2016003. Google Scholar
[24]	M. Keel and T. Tao, Endpoint strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039. Google Scholar
[25]	C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical, focusing, nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4. Google Scholar
[26]	J. Krieger, E. Lenzmann and P. Raphaël, On stability of pseudo-conformal blowup for $L^2$-critical Hartree NLS, Ann. Henri Poincaré, 10 (2009), 1159-1205. doi: 10.1007/s00023-009-0010-2. Google Scholar
[27]	M. K. Kwong, Uniqueness of positive solution of $\Delta u - u + u^p = 0$ in $ \mathbb R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502. Google Scholar
[28]	E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105. doi: 10.1002/sapm197757293. Google Scholar
[29]	C. Miao, G. Xu and L. Zhao, The cauchy problem of the hartree equation, J. Partial Diff. Eqs., 21 (2008), 22-44. Google Scholar
[30]	V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184. doi: 10.1016/j.jfa.2013.04.007. Google Scholar
[31]	T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330. doi: 10.1016/0022-0396(91)90052-B. Google Scholar
[32]	C. Sun, H. Wang, X. Yao and J. Zheng, Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 38 (2018), 2207-2228. doi: 10.3934/dcds.2018091. Google Scholar
[33]	E. Stein and T. S. Murphy, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton University Press, Princeton, 1993. Google Scholar
[34]	J. Stubbe, Global solutions and stable ground states of nonlinear Schrödinger equations, Phys. D, 48 (1991), 259-272. doi: 10.1016/0167-2789(91)90087-P. Google Scholar
[35]	W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517. Google Scholar
[36]	T. Tao, On the asymptotic behavior of large radial data for a focusing nonlinear Schrödinger equation, Dyn. Partial Differ. Equ., 1 (2004), 1-48. doi: 10.4310/DPDE.2004.v1.n1.a1. Google Scholar
[37]	T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343. doi: 10.1080/03605300701588805. Google Scholar
[38]	M. C. Vilela, Inhomogeneous strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., 359 (2007), 2123-2136. doi: 10.1090/S0002-9947-06-04099-2. Google Scholar
[39]	M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. Google Scholar
[40]	C. L. Xiang, Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions, Calc. Var. Partial Differential Equations, 55 (2016), Art. 134, 25 pp. doi: 10.1007/s00526-016-1068-6. Google Scholar
[41]	C. Xu and T. Zhao, A remark on the scattering theory for the 2D radial focusing INLS, preprint, arXiv: 1908.00743. Google Scholar

[1]	Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091
[2]	Jincai Kang, Chunlei Tang. Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5239-5252. doi: 10.3934/cpaa.2020235
[3]	Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395
[4]	Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure & Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015
[5]	Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003
[6]	Kenji Nakanishi, Tristan Roy. Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2023-2058. doi: 10.3934/cpaa.2016026
[7]	Kazuhiro Kurata, Tatsuya Watanabe. A remark on asymptotic profiles of radial solutions with a vortex to a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 597-610. doi: 10.3934/cpaa.2006.5.597
[8]	Alp Eden, Elİf Kuz. Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803
[9]	Qian Shen, Na Wei. Stability of ground state for the Schrödinger-Poisson equation. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020095
[10]	D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[11]	Dengfeng Lü. Existence and concentration behavior of ground state solutions for magnetic nonlinear Choquard equations. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1781-1795. doi: 10.3934/cpaa.2016014
[12]	Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259
[13]	Gui-Dong Li, Chun-Lei Tang. Existence of positive ground state solutions for Choquard equation with variable exponent growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2035-2050. doi: 10.3934/dcdss.2019131
[14]	Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[15]	Georgios Fotopoulos, Markus Harju, Valery Serov. Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D. Inverse Problems & Imaging, 2013, 7 (1) : 183-197. doi: 10.3934/ipi.2013.7.183
[16]	Satoshi Masaki. A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1481-1531. doi: 10.3934/cpaa.2015.14.1481
[17]	F. Catoire, W. M. Wang. Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori. Communications on Pure & Applied Analysis, 2010, 9 (2) : 483-491. doi: 10.3934/cpaa.2010.9.483
[18]	Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905
[19]	Chao Ji. Ground state solutions of fractional Schrödinger equations with potentials and weak monotonicity condition on the nonlinear term. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6071-6089. doi: 10.3934/dcdsb.2019131
[20]	Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807

American Institute of Mathematical Sciences

The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors