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

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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 and Continuous Dynamical Systems, 2019, 39 (9) : 5263-5273. doi: 10.3934/dcds.2019214

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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