### AIMS Mathematics

2021, Issue 6: 5837-5850. doi: 10.3934/math.2021345
Research article

# Existence of stable standing waves for the nonlinear Schrödinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities

• Received: 25 January 2021 Accepted: 17 March 2021 Published: 26 March 2021
• MSC : 35Q55

• In this paper, we investigate the existence of stable standing waves for the nonlinear Schrödinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities

$i \partial_t\psi+\triangle \psi+\frac{\gamma}{|x|^\alpha}\psi+\lambda_1|\psi|^p\psi +\lambda_2(I_\beta\ast|\psi|^q)|\psi|^{q-2}\psi = 0,\; \; (t,x)\in [0,T^\star)\times \mathbb{R}^N.$

By using concentration compactness principle, when one nonlinearity is focusing and $L^2$-critical, the other is defocusing and $L^2$-supercritical, we prove the existence and orbital stability of standing waves. We extend the results of Li-Zhao in paper [14] to the $L^2$-critical and $L^2$-supercritical nonlinearities.

Citation: Yile Wang. Existence of stable standing waves for the nonlinear Schrödinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities[J]. AIMS Mathematics, 2021, 6(6): 5837-5850. doi: 10.3934/math.2021345

### Related Papers:

• In this paper, we investigate the existence of stable standing waves for the nonlinear Schrödinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities

$i \partial_t\psi+\triangle \psi+\frac{\gamma}{|x|^\alpha}\psi+\lambda_1|\psi|^p\psi +\lambda_2(I_\beta\ast|\psi|^q)|\psi|^{q-2}\psi = 0,\; \; (t,x)\in [0,T^\star)\times \mathbb{R}^N.$

By using concentration compactness principle, when one nonlinearity is focusing and $L^2$-critical, the other is defocusing and $L^2$-supercritical, we prove the existence and orbital stability of standing waves. We extend the results of Li-Zhao in paper [14] to the $L^2$-critical and $L^2$-supercritical nonlinearities.

 [1] R. Y. Chiao, E. Garmire, C. H. Townes, Self-trapping of optical beams, Phys. Rev. Lett., 13 (1964), 479–482. doi: 10.1103/PhysRevLett.13.479 [2] J. P. Dong, Y. Lu, Infinite wall in the fractional quantum mechanics, J. Math. Phys., 62 (2021), 032104. doi: 10.1063/5.0026816 [3] L. R., Spectrum of atomic hydrogen: G. W. Series, Nuclear Physics, 6 (1958), 135–136. [4] C. Sulem, P. L. Sulem, The nonlinear Schrödinger equation: Self-focusing and wave collapse, New York: Springer, 1999. [5] V. E. Zakharov, Collapse of Langmuir waves, Sov. Phys. Jetp, 35 (1972), 908–914. [6] A. Bensouilah, V. D. Dinh, S. H. Zhu, On stability and instability of standing waves for the nonlinear Schrödinger equation with an inverse-square potential, J. Math. Phys., 59 (2018), 101505. doi: 10.1063/1.5038041 [7] B. H. Feng, H. H. Zhang, Stability of standing waves for the fractional Schrödinger-Choquard equation, Comput. Math. Appl., 75 (2018), 2499–2507. doi: 10.1016/j.camwa.2017.12.025 [8] B. H. Feng, R. P. Chen, Q. X. Wang, Instability of standing waves for the nonlinear Schrödinger-Poisson equation in the $L^2$-critical case, J. Dyn. Diff. Equat., 32 (2020), 1357–1370. doi: 10.1007/s10884-019-09779-6 [9] B. H. Feng, L. J. Cao, J. Y. Liu, Existence of stable standing waves for the Lee-Huang-Yang corrected dipolar Gross-Pitaevskii equation, Appl. Math. Lett., 115 (2021), 106952. doi: 10.1016/j.aml.2020.106952 [10] B. H. Feng, R. P. Chen, J. Y. Liu, Blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation, Adv. Nonlinear Anal., 10 (2021), 311–330. [11] T. Cazenave, P. L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Commun. Math. Phys., 85 (1982), 549–561. doi: 10.1007/BF01403504 [12] B. H. Feng, X. X. Yuan, On the Cauchy problem for the Schrödinger-Hartree equation, Evol. Equ. Control The., 4 (2015), 431–445. doi: 10.3934/eect.2015.4.431 [13] B. H. Feng, Q. X. Wang, Strong instability of standing waves for the nonlinear schrödinger equation in trapped dipolar quantum gases, J. Dyn. Diff. Equat., (2020). [14] X. F. Li, J. Y. Zhao, Orbital stability of standing waves for Schrödinger type equations with slowly decaying linear potential, Comput. Math. Appl., 79 (2020), 303–316. doi: 10.1016/j.camwa.2019.06.030 [15] Z. P. Cheng, M. B. Yang, Stability of standing waves for a generalized Choquard equation with potential, Acta Appl. Math., 157 (2018), 25–44. doi: 10.1007/s10440-018-0162-5 [16] X. Luo, H. Y. Ye, Multiplicity and stability of standing waves for the nonlinear Schrödinger-Poisson equation with a harmonic potential, Math. Methods Appl. Sci., 42(2019), 1844–1858. doi: 10.1002/mma.5478 [17] T. Cazenave, Semilinear Schrödinger Equations, New York: American Mathematical Society, 2003. [18] E. H. Lieb, Analysis, 2 Eds., Graduate Studies in Mathematics, 14 (1997). [19] B. H. Feng, R. P. Chen, J. J. Ren, Existence of stable standing waves for the fractional Schrödinger equations with combined power-type and Choquard-type nonlinearities, J. Math. Phys., 60 (2019), 051512. doi: 10.1063/1.5082684 [20] V. Moroz, J. V. 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
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