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September  2019, 39(9): 5263-5273. doi: 10.3934/dcds.2019214

Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation

Xiaoyu Zeng and  Yimin Zhang ,

Center for Mathematical Sciences and Department of Mathematics, Wuhan University of Technology, Wuhan 430070, China

* Corresponding author: Yimin Zhang

Received  September 2018 Revised  March 2019 Published  May 2019

Fund Project: The first author is supported by NSFC grants 11601173, 11871387 and the Fundamental Research Funds for the Central Universities(WUT: 2017 IVA 076). The second author is supported by NSFC grants 11671394, 11771127 and the Fundamental Research Funds for the Central Universities (WUT: 2018IB014).

In this paper, we consider $ L^2 $ constrained minimization problem for a modified Gross-Pitaevskii equation with higher order interactions in $ \mathbb{R}^2 $. By using an auxiliary functional and some detailed energy estimates, the blow-up behavior of ground state for the modified Gross-Pitaevskii equation was obtained under different parameter regimes. Our conclusion extends some results of [3,Theorem 3.4].

Keywords: $ L^2 $-normalized solutions, modified Gross-Pitaevskii equation, asymptotic behavior.
Mathematics Subject Classification: 35J20, 35J60.
Citation: Xiaoyu Zeng, Yimin Zhang. Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5263-5273. doi: 10.3934/dcds.2019214
References:
[1]

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W. Z. Bao, Y. Y. Cai and X. R. Ruan, Ground states of Bose-Einstein condensation with higher order interaction, Physica D, 386/387 (2019), 38-48.  doi: 10.1016/j.physd.2018.08.006.  Google Scholar

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Y. J. Guo, Z.-Q. Wang, X. Y. Zeng and H. S. Zhou, Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials, Nonlinearity, 31 (2018), 957-979.  doi: 10.1088/1361-6544/aa99a8.  Google Scholar

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Y. J. Guo, X. Y. Zeng and H. S. Zhou, Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2014), 809-828.  doi: 10.1016/j.anihpc.2015.01.005.  Google Scholar

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M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $\mathbb{R}^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

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Z. X. Li and Y. M. Zhang, Solutions for a class of quasilinear Schrödinger equations with critical Sobolev exponents, J. Math. Phys., 58 (2017), 021501, 15pp. doi: 10.1063/1.4975009.  Google Scholar

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J. Q. Liu, Y. Q. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differential Equations, 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

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J.Q. Liu, Y.Q. Wang and Z.-Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901.  doi: 10.1081/PDE-120037335.  Google Scholar

[17]

X. R. Ruan, Y. Y. Cai and W. Z. Bao, Mean-field regime Thomas-Fermi approximations of trapped Bose-Einstein condensates with higher order interactions in one and two dimensions, J. Physics B Atomic, 82 (2015), 109-114.   Google Scholar

[18]

H. Veksler, S. Fishman and W. Ketterle, Simple model for interactions and corrections to the Gross-Pitaevskii equation, Phys. Rev. A., 90 (2014), 023620. doi: 10.1103/PhysRevA.90.023620.  Google Scholar

[19]

X. Y. Zeng, Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 37 (2017), 1749-1762.  doi: 10.3934/dcds.2017073.  Google Scholar

[20]

X. Y. Zeng and Y. M. Zhang, Existence and uniqueness of normalized solutions for the Kirchhoff equation, Applied Math. Lett., 74 (2017), 52-59.  doi: 10.1016/j.aml.2017.05.012.  Google Scholar

[21]

X. Y. Zeng and Y. M. Zhang, Existence and asymptotic behavior for the ground state of quasilinear elliptic equations, Adv. Nonlinear Stud., 18 (2018), 725-744.  doi: 10.1515/ans-2018-0005.  Google Scholar

show all references

References:
[1]

M. Agueh, Sharp Gagliardo-Nirenberg Inequalities via p-Laplacian type equations, Nonlinear Differ. Equ. Appl., 15 (2008), 457-472.  doi: 10.1007/s00030-008-7021-4.  Google Scholar

[2]

W. Z. Bao and Y. Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.  Google Scholar

[3]

W. Z. Bao, Y. Y. Cai and X. R. Ruan, Ground states of Bose-Einstein condensation with higher order interaction, Physica D, 386/387 (2019), 38-48.  doi: 10.1016/j.physd.2018.08.006.  Google Scholar

[4]

A. Collin, P. Massignan and C. J. Pethik, Energy-dependent effective interactions for dilute many-body systems, Phys. Rev. A., 75 (2007), 013615. doi: 10.1103/PhysRevA.75.013615.  Google Scholar

[5]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach, Nonlinear Anal., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.  Google Scholar

[6]

M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23 (2010), 1353-1385.  doi: 10.1088/0951-7715/23/6/006.  Google Scholar

[7]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^n$, Mathematical analysis and applications Part A, Adv. in Math. Suppl. Stud., 7a (1981), 369-402.   Google Scholar

[8]

Y. J. Guo and R. Seiringer, On the mass concentration for Bose-Einstein condensates with attractive interactions, Lett. Math. Phys., 104 (2014), 141-156.  doi: 10.1007/s11005-013-0667-9.  Google Scholar

[9]

Y. J. Guo, Z.-Q. Wang, X. Y. Zeng and H. S. Zhou, Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials, Nonlinearity, 31 (2018), 957-979.  doi: 10.1088/1361-6544/aa99a8.  Google Scholar

[10]

Y. J. Guo, X. Y. Zeng and H. S. Zhou, Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2014), 809-828.  doi: 10.1016/j.anihpc.2015.01.005.  Google Scholar

[11]

L. Jeanjean and T. J. Luo, Sharp nonexistence results of prescribed $L^2$-norm solutions for some class of Schrödinger-Poisson and quasi-linear equations, Zeitschrift für angewandte Math. und Physik, 64 (2013), 937-954.  doi: 10.1007/s00033-012-0272-2.  Google Scholar

[12]

L. Jeanjean, T. J. 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

[13]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $\mathbb{R}^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[14]

Z. X. Li and Y. M. Zhang, Solutions for a class of quasilinear Schrödinger equations with critical Sobolev exponents, J. Math. Phys., 58 (2017), 021501, 15pp. doi: 10.1063/1.4975009.  Google Scholar

[15]

J. Q. Liu, Y. Q. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differential Equations, 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[16]

J.Q. Liu, Y.Q. Wang and Z.-Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901.  doi: 10.1081/PDE-120037335.  Google Scholar

[17]

X. R. Ruan, Y. Y. Cai and W. Z. Bao, Mean-field regime Thomas-Fermi approximations of trapped Bose-Einstein condensates with higher order interactions in one and two dimensions, J. Physics B Atomic, 82 (2015), 109-114.   Google Scholar

[18]

H. Veksler, S. Fishman and W. Ketterle, Simple model for interactions and corrections to the Gross-Pitaevskii equation, Phys. Rev. A., 90 (2014), 023620. doi: 10.1103/PhysRevA.90.023620.  Google Scholar

[19]

X. Y. Zeng, Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 37 (2017), 1749-1762.  doi: 10.3934/dcds.2017073.  Google Scholar

[20]

X. Y. Zeng and Y. M. Zhang, Existence and uniqueness of normalized solutions for the Kirchhoff equation, Applied Math. Lett., 74 (2017), 52-59.  doi: 10.1016/j.aml.2017.05.012.  Google Scholar

[21]

X. Y. Zeng and Y. M. Zhang, Existence and asymptotic behavior for the ground state of quasilinear elliptic equations, Adv. Nonlinear Stud., 18 (2018), 725-744.  doi: 10.1515/ans-2018-0005.  Google Scholar

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