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Asymptotic spreading speed for the weak competition system with a free boundary
Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation
Center for Mathematical Sciences and Department of Mathematics, Wuhan University of Technology, Wuhan 430070, China |
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 [
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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