-
Previous Article
The bifurcations of solitary and kink waves described by the Gardner equation
- DCDS-S Home
- This Issue
-
Next Article
Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection
Sharp variational characterization and a Schrödinger equation with Hartree type nonlinearity
1. | School of Mathematics and Computer Science, Fujian Normal University, Qishan Campus, Fuzhou 350117, China |
References:
[1] |
P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation, Adv. Math. Sci. Appl., 12 (2002), 817-827. |
[2] |
Y. Cao, Z. H. Musslimani and E. S. Titi, Nonlinear Schrödinger -Helmholtz equation as numerical regularization of the nonlinear Schrödinger equation, Nonlinearity, 21 (2008), 879-898.
doi: 10.1088/0951-7715/21/5/001. |
[3] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Institute of Mathematical Sciences, Vol. 10, Providence, Rhode Island, 2005.
doi: 10.1090/cln/010. |
[4] |
T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85 (1982), 549-561.
doi: 10.1007/BF01403504. |
[5] |
G. Chen and J. Zhang, Remarks on global esistence for the supercritical nonlinear Schrödinger equation with a harmonic potential, J. Math. Anal. Appl., 320 (2006), 591-598.
doi: 10.1016/j.jmaa.2005.07.008. |
[6] |
J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation, Phys. D, 227 (2007), 142-148.
doi: 10.1016/j.physd.2007.01.004. |
[7] |
J. Chen, B. Guo and Y. Han, Sharp constant in nonlocal inequality and its applications to nonlocal Schrödinger equation with harmonic potential, Commun. Math. Sci., 7 (2009), 549-570.
doi: 10.4310/CMS.2009.v7.n3.a2. |
[8] |
J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equation, I: The Cauchy problem, J. Funct. Anal., 32 (1979), 33-71.
doi: 10.1016/0022-1236(79)90077-6. |
[9] |
R. T. Glassey, On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. Math. Phys., 18 (1977), 1794-1797.
doi: 10.1063/1.523491. |
[10] |
E. P. Gross, Physics of many-particle systems, (eds. E. Meeron), New York-London-Paris, Gordon Breash, 1 (1966), 231-406. |
[11] |
P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part 1 and 2, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 109-145 and 223-283. |
[12] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vols. II, IV, Elsevier (Singapore) Pte Ltd, 2003. |
[13] |
M. Kurth, On the existence of infinitely many modes of a nonlocal nonlinear Schrödinger equation related to Dispersion-Managed solitons, SIAM J. Math. Anal., 36 (2004), 967-985.
doi: 10.1137/S0036141003431530. |
[14] |
M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. |
[15] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[16] |
J. Zhang, Stability of attractive Bose-Einstein condensates, J. Statistical Phys., 101 (2000), 731-746.
doi: 10.1023/A:1026437923987. |
show all references
References:
[1] |
P. Bégout, Necessary conditions and sufficient conditions for global existence in the nonlinear Schrödinger equation, Adv. Math. Sci. Appl., 12 (2002), 817-827. |
[2] |
Y. Cao, Z. H. Musslimani and E. S. Titi, Nonlinear Schrödinger -Helmholtz equation as numerical regularization of the nonlinear Schrödinger equation, Nonlinearity, 21 (2008), 879-898.
doi: 10.1088/0951-7715/21/5/001. |
[3] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Institute of Mathematical Sciences, Vol. 10, Providence, Rhode Island, 2005.
doi: 10.1090/cln/010. |
[4] |
T. Cazenave and P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85 (1982), 549-561.
doi: 10.1007/BF01403504. |
[5] |
G. Chen and J. Zhang, Remarks on global esistence for the supercritical nonlinear Schrödinger equation with a harmonic potential, J. Math. Anal. Appl., 320 (2006), 591-598.
doi: 10.1016/j.jmaa.2005.07.008. |
[6] |
J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation, Phys. D, 227 (2007), 142-148.
doi: 10.1016/j.physd.2007.01.004. |
[7] |
J. Chen, B. Guo and Y. Han, Sharp constant in nonlocal inequality and its applications to nonlocal Schrödinger equation with harmonic potential, Commun. Math. Sci., 7 (2009), 549-570.
doi: 10.4310/CMS.2009.v7.n3.a2. |
[8] |
J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equation, I: The Cauchy problem, J. Funct. Anal., 32 (1979), 33-71.
doi: 10.1016/0022-1236(79)90077-6. |
[9] |
R. T. Glassey, On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. Math. Phys., 18 (1977), 1794-1797.
doi: 10.1063/1.523491. |
[10] |
E. P. Gross, Physics of many-particle systems, (eds. E. Meeron), New York-London-Paris, Gordon Breash, 1 (1966), 231-406. |
[11] |
P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part 1 and 2, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 109-145 and 223-283. |
[12] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vols. II, IV, Elsevier (Singapore) Pte Ltd, 2003. |
[13] |
M. Kurth, On the existence of infinitely many modes of a nonlocal nonlinear Schrödinger equation related to Dispersion-Managed solitons, SIAM J. Math. Anal., 36 (2004), 967-985.
doi: 10.1137/S0036141003431530. |
[14] |
M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. |
[15] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[16] |
J. Zhang, Stability of attractive Bose-Einstein condensates, J. Statistical Phys., 101 (2000), 731-746.
doi: 10.1023/A:1026437923987. |
[1] |
Maria J. Esteban. Gagliardo-Nirenberg-Sobolev inequalities on planar graphs. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2101-2114. doi: 10.3934/cpaa.2022051 |
[2] |
Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure and Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011 |
[3] |
Wenyan Zhang, Shu Xu, Shengji Li, Xuexiang Huang. Generalized weak sharp minima of variational inequality problems with functional constraints. Journal of Industrial and Management Optimization, 2013, 9 (3) : 621-630. doi: 10.3934/jimo.2013.9.621 |
[4] |
Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134 |
[5] |
Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101 |
[6] |
YanYan Li, Tonia Ricciardi. A sharp Sobolev inequality on Riemannian manifolds. Communications on Pure and Applied Analysis, 2003, 2 (1) : 1-31. doi: 10.3934/cpaa.2003.2.1 |
[7] |
C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial and Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519 |
[8] |
Abd-semii Oluwatosin-Enitan Owolabi, Timilehin Opeyemi Alakoya, Adeolu Taiwo, Oluwatosin Temitope Mewomo. A new inertial-projection algorithm for approximating common solution of variational inequality and fixed point problems of multivalued mappings. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 255-278. doi: 10.3934/naco.2021004 |
[9] |
Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3407-3438. doi: 10.3934/dcds.2018146 |
[10] |
Micol Amar, Andrea Braides. A characterization of variational convergence for segmentation problems. Discrete and Continuous Dynamical Systems, 1995, 1 (3) : 347-369. doi: 10.3934/dcds.1995.1.347 |
[11] |
Rodolfo Ríos-Zertuche. Characterization of minimizable Lagrangian action functionals and a dual Mather theorem. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2615-2639. doi: 10.3934/dcds.2020143 |
[12] |
James Nolen, Jack Xin. Existence of KPP type fronts in space-time periodic shear flows and a study of minimal speeds based on variational principle. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1217-1234. doi: 10.3934/dcds.2005.13.1217 |
[13] |
Gianira N. Alfarano, Martino Borello, Alessandro Neri. A geometric characterization of minimal codes and their asymptotic performance. Advances in Mathematics of Communications, 2022, 16 (1) : 115-133. doi: 10.3934/amc.2020104 |
[14] |
Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437 |
[15] |
Guofeng Che, Haibo Chen, Tsung-fang Wu. Bound state positive solutions for a class of elliptic system with Hartree nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3697-3722. doi: 10.3934/cpaa.2020163 |
[16] |
Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106 |
[17] |
Purshottam Narain Agrawal, Sompal Singh. Stancu variant of Jakimovski-Leviatan-Durrmeyer operators involving Brenke type polynomials. Mathematical Foundations of Computing, 2022 doi: 10.3934/mfc.2022004 |
[18] |
Parveen Bawa, Neha Bhardwaj, P. N. Agrawal. Quantitative Voronovskaya type theorems and GBS operators of Kantorovich variant of Lupaş-Stancu operators based on Pólya distribution. Mathematical Foundations of Computing, 2022 doi: 10.3934/mfc.2022003 |
[19] |
Jeremy Levesley, Xinping Sun, Fahd Jarad, Alexander Kushpel. Interpolation of exponential-type functions on a uniform grid by shifts of a basis function. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2399-2416. doi: 10.3934/dcdss.2020403 |
[20] |
Zijia Peng, Cuiming Ma, Zhonghui Liu. Existence for a quasistatic variational-hemivariational inequality. Evolution Equations and Control Theory, 2020, 9 (4) : 1153-1165. doi: 10.3934/eect.2020058 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]