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Normalized solutions for Choquard equations with general nonlinearities
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China |
$ L^{2} $ |
$ \begin{equation*} -\Delta u-\lambda u = (I_{\alpha}*F(u))f(u), \ \ \ \ x\in \mathbb{R}^3, \end{equation*} $ |
$ \lambda\in \mathbb{R}, \alpha\in (0,3) $ |
$ I_{\alpha}: \mathbb{R}^3\rightarrow \mathbb{R} $ |
$ c>0 $ |
$ (\bar{u}_c, \bar{ \lambda}_c)\in \mathcal{S}_{c}\times \mathbb{R}^- $ |
$ \|\bar{u}_c\|_{2}^{2} = c $ |
References:
[1] |
T. Bartsch, L. Jeanjean and N. Soave,
Normalized solutions for a system of coupled cubic Schrödinger equations on $\Bbb{R}^3$, J. Math. Pures Appl., 106 (2016), 583-614.
doi: 10.1016/j.matpur.2016.03.004. |
[2] |
T. Bartsch and N. Soave,
A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, J. Funct. Anal., 272 (2017), 4998-5037.
doi: 10.1016/j.jfa.2017.01.025. |
[3] |
J. Bellazzini and G. Siciliano,
Scaling properties of functionals and existence of constrained minimizers, J. Funct. Anal., 261 (2011), 2486-2507.
doi: 10.1016/j.jfa.2011.06.014. |
[4] |
J. Bellazzini, L. Jeanjean and 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.
doi: 10.1112/plms/pds072. |
[5] |
D. Cao and H. Li,
High energy solutions of the Choquard equation, Discrete Contin. Dyn. Syst., 38 (2018), 3023-3032.
doi: 10.3934/dcds.2018129. |
[6] |
S. Chen, J. Shi and X. Tang,
Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 5867-5889.
doi: 10.3934/dcds.2019257. |
[7] |
S. Chen and X. Tang,
Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials, Adv. Nonlinear Anal., 9 (2020), 496-515.
doi: 10.1515/anona-2020-0011. |
[8] |
S. Chen, X. Tang and S. Yuan, Normalized solutions for Schrödinger-Poisson equations with general nonlinearities, J. Math. Anal. Appl., 481 (2020), 123447, 24 pp.
doi: 10.1016/j.jmaa.2019.123447. |
[9] |
S. Chen and X. Tang,
On the planar Schrödinger-Poisson system with the axially symmetric potential, J. Differential Equations, 268 (2020), 945-976.
doi: 10.1016/j.jde.2019.08.036. |
[10] |
S. Chen, A. Fiscella, P. Pucci and X. Tang,
Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations, J. Differential Equations, 268 (2020), 2672-2716.
doi: 10.1016/j.jde.2019.09.041. |
[11] |
P. Choquard, J. Stubbe and M. Vuffray,
Stationary solutions of the Schrödinger-Newton model–an ODE approach, Differential Integral Equations, 21 (2008), 665-679.
|
[12] |
L. Jeanjean,
Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659.
doi: 10.1016/S0362-546X(96)00021-1. |
[13] |
L. Jeanjean and T. Luo,
Sharp nonexistence results of prescribed $L^2$-norm solutions for some class of Schrödinger-Poisson and quasi-linear equations, Z. Angew. Math. Phys., 64 (2013), 937-954.
doi: 10.1007/s00033-012-0272-2. |
[14] |
L. Jeanjean, T. Luo and Z.-Q. Wang,
Multiple normalized solutions for quasi-linear Schrödinger equations, J. Differential Equations, 259 (2015), 3894-3928.
doi: 10.1016/j.jde.2015.05.008. |
[15] |
Y. Lei,
On finite energy solutions of fractional order equations of the Choquard type, Discrete Contin. Dyn. Syst., 39 (2019), 1497-1515.
doi: 10.3934/dcds.2019064. |
[16] |
G.-B. Li and H.-Y. Ye, The existence of positive solutions with prescribed $L^2$-norm for nonlinear Choquard equations, J. Math. Phys., 55 (2014), 19 pp.
doi: 10.1063/1.4902386. |
[17] |
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. |
[18] |
P.-L. Lions,
Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97.
doi: 10.1007/BF01205672. |
[19] |
G. P. Menzala,
On regular solutions of a nonlinear equation of Choquard's type, Proc. Roy. Soc. Edinburgh Sect. A, 86 (1980), 291-301.
doi: 10.1017/S0308210500012191. |
[20] |
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. |
[21] |
I. M. Moroz, R. Penrose and P. Tod,
Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.
doi: 10.1088/0264-9381/15/9/019. |
[22] |
S. I. Pekar, Üntersuchung über die Elektronentheorie der Kristalle, Akademie-Verlag, 1954. Google Scholar |
[23] |
R. Penrose,
On gravity's role in quantum state reduction, Gen. Relativity Gravitation, 28 (1996), 581-600.
doi: 10.1007/BF02105068. |
[24] |
X. Tang and S. Chen,
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Discrete Contin. Dyn. Syst., 37 (2017), 4973-5002.
doi: 10.3934/dcds.2017214. |
[25] |
X. Tang and S. Chen,
Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions, Adv. Nonlinear Anal., 9 (2020), 413-437.
doi: 10.1515/anona-2020-0007. |
[26] |
X. Tang, S. Chen, X. Lin and J. Yu, Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions, J. Differential Equations, 268 (2019).
doi: 10.1016/j.jde.2019.10.041. |
[27] |
P. Tod and I. M. Moroz,
An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12 (1999), 201-216.
doi: 10.1088/0951-7715/12/2/002. |
[28] |
J. Vétois and S. Wang,
Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four, Adv. Nonlinear Anal., 8 (2019), 715-724.
doi: 10.1515/anona-2017-0085. |
[29] |
M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[30] |
H. Ye, The mass concentration phenomenon for $L^2$-critical constrained problems related to Kirchhoff equations, Z. Angew. Math. Phys., 67 (2016), 16 pp.
doi: 10.1007/s00033-016-0624-4. |
show all references
References:
[1] |
T. Bartsch, L. Jeanjean and N. Soave,
Normalized solutions for a system of coupled cubic Schrödinger equations on $\Bbb{R}^3$, J. Math. Pures Appl., 106 (2016), 583-614.
doi: 10.1016/j.matpur.2016.03.004. |
[2] |
T. Bartsch and N. Soave,
A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, J. Funct. Anal., 272 (2017), 4998-5037.
doi: 10.1016/j.jfa.2017.01.025. |
[3] |
J. Bellazzini and G. Siciliano,
Scaling properties of functionals and existence of constrained minimizers, J. Funct. Anal., 261 (2011), 2486-2507.
doi: 10.1016/j.jfa.2011.06.014. |
[4] |
J. Bellazzini, L. Jeanjean and 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.
doi: 10.1112/plms/pds072. |
[5] |
D. Cao and H. Li,
High energy solutions of the Choquard equation, Discrete Contin. Dyn. Syst., 38 (2018), 3023-3032.
doi: 10.3934/dcds.2018129. |
[6] |
S. Chen, J. Shi and X. Tang,
Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity, Discrete Contin. Dyn. Syst., 39 (2019), 5867-5889.
doi: 10.3934/dcds.2019257. |
[7] |
S. Chen and X. Tang,
Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials, Adv. Nonlinear Anal., 9 (2020), 496-515.
doi: 10.1515/anona-2020-0011. |
[8] |
S. Chen, X. Tang and S. Yuan, Normalized solutions for Schrödinger-Poisson equations with general nonlinearities, J. Math. Anal. Appl., 481 (2020), 123447, 24 pp.
doi: 10.1016/j.jmaa.2019.123447. |
[9] |
S. Chen and X. Tang,
On the planar Schrödinger-Poisson system with the axially symmetric potential, J. Differential Equations, 268 (2020), 945-976.
doi: 10.1016/j.jde.2019.08.036. |
[10] |
S. Chen, A. Fiscella, P. Pucci and X. Tang,
Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations, J. Differential Equations, 268 (2020), 2672-2716.
doi: 10.1016/j.jde.2019.09.041. |
[11] |
P. Choquard, J. Stubbe and M. Vuffray,
Stationary solutions of the Schrödinger-Newton model–an ODE approach, Differential Integral Equations, 21 (2008), 665-679.
|
[12] |
L. Jeanjean,
Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659.
doi: 10.1016/S0362-546X(96)00021-1. |
[13] |
L. Jeanjean and T. Luo,
Sharp nonexistence results of prescribed $L^2$-norm solutions for some class of Schrödinger-Poisson and quasi-linear equations, Z. Angew. Math. Phys., 64 (2013), 937-954.
doi: 10.1007/s00033-012-0272-2. |
[14] |
L. Jeanjean, T. Luo and Z.-Q. Wang,
Multiple normalized solutions for quasi-linear Schrödinger equations, J. Differential Equations, 259 (2015), 3894-3928.
doi: 10.1016/j.jde.2015.05.008. |
[15] |
Y. Lei,
On finite energy solutions of fractional order equations of the Choquard type, Discrete Contin. Dyn. Syst., 39 (2019), 1497-1515.
doi: 10.3934/dcds.2019064. |
[16] |
G.-B. Li and H.-Y. Ye, The existence of positive solutions with prescribed $L^2$-norm for nonlinear Choquard equations, J. Math. Phys., 55 (2014), 19 pp.
doi: 10.1063/1.4902386. |
[17] |
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. |
[18] |
P.-L. Lions,
Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97.
doi: 10.1007/BF01205672. |
[19] |
G. P. Menzala,
On regular solutions of a nonlinear equation of Choquard's type, Proc. Roy. Soc. Edinburgh Sect. A, 86 (1980), 291-301.
doi: 10.1017/S0308210500012191. |
[20] |
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. |
[21] |
I. M. Moroz, R. Penrose and P. Tod,
Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.
doi: 10.1088/0264-9381/15/9/019. |
[22] |
S. I. Pekar, Üntersuchung über die Elektronentheorie der Kristalle, Akademie-Verlag, 1954. Google Scholar |
[23] |
R. Penrose,
On gravity's role in quantum state reduction, Gen. Relativity Gravitation, 28 (1996), 581-600.
doi: 10.1007/BF02105068. |
[24] |
X. Tang and S. Chen,
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Discrete Contin. Dyn. Syst., 37 (2017), 4973-5002.
doi: 10.3934/dcds.2017214. |
[25] |
X. Tang and S. Chen,
Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions, Adv. Nonlinear Anal., 9 (2020), 413-437.
doi: 10.1515/anona-2020-0007. |
[26] |
X. Tang, S. Chen, X. Lin and J. Yu, Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions, J. Differential Equations, 268 (2019).
doi: 10.1016/j.jde.2019.10.041. |
[27] |
P. Tod and I. M. Moroz,
An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12 (1999), 201-216.
doi: 10.1088/0951-7715/12/2/002. |
[28] |
J. Vétois and S. Wang,
Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four, Adv. Nonlinear Anal., 8 (2019), 715-724.
doi: 10.1515/anona-2017-0085. |
[29] |
M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[30] |
H. Ye, The mass concentration phenomenon for $L^2$-critical constrained problems related to Kirchhoff equations, Z. Angew. Math. Phys., 67 (2016), 16 pp.
doi: 10.1007/s00033-016-0624-4. |
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