### Electronic Research Archive

2020, Issue 1: 291-309. doi: 10.3934/era.2020017
Special Issues

# Normalized solutions for Choquard equations with general nonlinearities

• Received: 01 November 2019 Revised: 01 January 2020
• Primary: 58F15, 58F17; Secondary: 53C35

• 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 $\|\bar{u}_c\|_{2}^{2} = c$.

Citation: Shuai Yuan, Sitong Chen, Xianhua Tang. Normalized solutions for Choquard equations with general nonlinearities[J]. Electronic Research Archive, 2020, 28(1): 291-309. doi: 10.3934/era.2020017

### Related Papers:

• 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 $\|\bar{u}_c\|_{2}^{2} = c$.

 [1] Normalized solutions for a system of coupled cubic Schrödinger equations on $\Bbb{R}^3$. J. Math. Pures Appl. (2016) 106: 583-614. [2] A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems. J. Funct. Anal. (2017) 272: 4998-5037. [3] Scaling properties of functionals and existence of constrained minimizers. J. Funct. Anal. (2011) 261: 2486-2507. [4] Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations. Proc. Lond. Math. Soc. (2013) 107: 303-339. [5] High energy solutions of the Choquard equation. Discrete Contin. Dyn. Syst. (2018) 38: 3023-3032. [6] Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity. Discrete Contin. Dyn. Syst. (2019) 39: 5867-5889. [7] Berestycki-Lions conditions on ground state solutions for a Nonlinear Schrödinger equation with variable potentials. Adv. Nonlinear Anal. (2020) 9: 496-515. [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] On the planar Schrödinger-Poisson system with the axially symmetric potential. J. Differential Equations (2020) 268: 945-976. [10] Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations. J. Differential Equations (2020) 268: 2672-2716. [11] Stationary solutions of the Schrödinger-Newton model–an ODE approach. Differential Integral Equations (2008) 21: 665-679. [12] Existence of solutions with prescribed norm for semilinear elliptic equations. Nonlinear Anal. (1997) 28: 1633-1659. [13] Sharp nonexistence results of prescribed $L^2$-norm solutions for some class of Schrödinger-Poisson and quasi-linear equations. Z. Angew. Math. Phys. (2013) 64: 937-954. [14] Multiple normalized solutions for quasi-linear Schrödinger equations. J. Differential Equations (2015) 259: 3894-3928. [15] On finite energy solutions of fractional order equations of the Choquard type. Discrete Contin. Dyn. Syst. (2019) 39: 1497-1515. [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] Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Studies in Appl. Math. (1977) 57: 93-105. [18] Solutions of Hartree-Fock equations for Coulomb systems. Comm. Math. Phys. (1987) 109: 33-97. [19] On regular solutions of a nonlinear equation of Choquard's type. Proc. Roy. Soc. Edinburgh Sect. A (1980) 86: 291-301. [20] Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics. J. Funct. Anal. (2013) 265: 153-184. [21] Spherically-symmetric solutions of the Schrödinger-Newton equations. Classical Quantum Gravity (1998) 15: 2733-2742. [22] S. I. Pekar, Üntersuchung über die Elektronentheorie der Kristalle, Akademie-Verlag, 1954. [23] On gravity's role in quantum state reduction. Gen. Relativity Gravitation (1996) 28: 581-600. [24] Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials. Discrete Contin. Dyn. Syst. (2017) 37: 4973-5002. [25] Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions. Adv. Nonlinear Anal. (2020) 9: 413-437. [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] An analytical approach to the Schrödinger-Newton equations. Nonlinearity (1999) 12: 201-216. [28] Infinitely many solutions for cubic nonlinear Schrödinger equations in dimension four. Adv. Nonlinear Anal. (2019) 8: 715-724. [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
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.833 0.8