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March  2020, 28(1): 291-309. doi: 10.3934/era.2020017

Normalized solutions for Choquard equations with general nonlinearities

Shuai Yuan , Sitong Chen , and  Xianhua Tang

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

* Corresponding author: Sitong Chen

Received  November 2019 Revised  January 2020 Published  March 2020

Fund Project: This work was partially supported by the National Natural Science Foundation of China (No: 1197011711)

In this paper, we prove the existence of positive solutions with prescribed
$ L^{2} $
-norm to the following Choquard equation:
$ \begin{equation*} -\Delta u-\lambda u = (I_{\alpha}*F(u))f(u), \ \ \ \ x\in \mathbb{R}^3, \end{equation*} $
where
$ \lambda\in \mathbb{R}, \alpha\in (0,3) $
 and
$ I_{\alpha}: \mathbb{R}^3\rightarrow \mathbb{R} $
 is the Riesz potential. Under the weaker conditions, by using a minimax procedure and some new analytical techniques, we show that for any
$ c>0 $
, the above equation possesses at least a couple of weak solution
$ (\bar{u}_c, \bar{ \lambda}_c)\in \mathcal{S}_{c}\times \mathbb{R}^- $
 such that
$ \|\bar{u}_c\|_{2}^{2} = c $
.
Keywords: Choquard equations, normalized solution, variational method, minimax method, weak solutions.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Shuai Yuan, Sitong Chen, Xianhua Tang. Normalized solutions for Choquard equations with general nonlinearities. Electronic Research Archive, 2020, 28 (1) : 291-309. doi: 10.3934/era.2020017
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97.  doi: 10.1007/BF01205672.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97.  doi: 10.1007/BF01205672.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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.  Google Scholar

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