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January  2019, 18(1): 65-81. doi: 10.3934/cpaa.2019005

Constraint minimizers of perturbed gross-pitaevskii energy functionals in $\mathbb{R}^N$

Shuai Li 1, , Jingjing Yan 2, and  Xincai Zhu 3,,

1. 

College of Science, Huazhong Agricultural University, Wuhan, 430070 Hubei, China
2. 

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, 430071 Hubei, China
3. 

School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000 Henan, China

* Corresponding author

Received  August 2017 Revised  January 2018 Published  August 2018

Fund Project: This research was partly supported by NSFC grant 11671394

This paper is concerned with constraint minimizers of an $L^2-$critical minimization problem (1) in $\mathbb{R}^N$ ($N≥ 1$) under an $L^2-$subcritical perturbation. We prove that the problem admits minimizers with mass $ρ^\frac{N}{2}$ if and only if $0≤ρ < ρ^*: = \|Q\|^{\frac{4}{N}}_2 $ for $b≥0$ and $0 < ρ ≤ρ^*$ for $b < 0$, where the constant $b$ comes from the coefficient of the perturbation term, and $Q$ is the unique positive radically symmetric solution of $Δ u(x)-u(x)+u^{1+\frac{4}{N}}(x) = 0$ in $\mathbb{R}^N$. Furthermore, we analyze rigorously the concentration behavior of minimizers as $ρ \nearrow ρ^*$ for the case where $b>0$, which shows that the concentration rates are determined by the subcritical perturbation, instead of the local profiles of the potential $V(x)$.

Keywords: Gross-Pitaevskii functional, subcritical perturbation, minimizers, energy estimate, mass concentration.
Mathematics Subject Classification: Primary: 35J20, 35J60, 35Q40.
Citation: Shuai Li, Jingjing Yan, Xincai Zhu. Constraint minimizers of perturbed gross-pitaevskii energy functionals in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2019, 18 (1) : 65-81. doi: 10.3934/cpaa.2019005
References:
[1]

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

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E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cimento, 20 (1961), 454-466.   Google Scholar

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E. P. Gross, Hydrodynamics of a superfluid condensate, J. Math. Phys., 4 (1963), 195-207.  doi: 10.1063/1.1703944.  Google Scholar

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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

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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, 30 (2016), 809-828.  doi: 10.1016/j.anihpc.2015.01.005.  Google Scholar

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S. Li, J. L. Xiang and X. Y. Zeng, Ground states of nonlinear Choquard equations with multi-well potentials, J. Math. Phys., 57 (2016), 081515. doi: 10.1063/1.4961158.  Google Scholar

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Y. Li and W. M. Ni, Radial symmetry of positive solutions of nonlinear elliptic equation in $\mathbb{R}^n$, Comm. Partial Differential Equations, 18 (1993), 1043-1054.  doi: 10.1080/03605309308820960.  Google Scholar

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P. L. Lions, The concentration-compactness principle in the caclulus of variations. The locally compact case. Ⅰ, Ann. Inst H. Poincaré. Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[19]

P. L. Lions, The concentration-compactness principle in the caclulus of variations. The locally compact case. Ⅱ, Ann. Inst H. Poincaré. Anal. Non Linéaire, 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[20]

L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Sov. Phys. JETP, 13 (1961), 451-454.  doi: 10.11572/8649.  Google Scholar

[21]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York-London, 1978. Google Scholar

[22]

X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  doi: 10.1007/BF02096642.  Google Scholar

[23]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolations estimates, Comm. Math. Phys., 87 (1983), 567-576.  doi: 10.1007/BF01208265.  Google Scholar

[24]

X. Y. Zeng and Y. M. Zhang, Existence and asymptotic behavior for the ground state of quasilinear elliptic equation, preprint arXiv: 1703.00183 Google Scholar

show all references

References:
[1]

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

[2]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N$, Comm. Partial Differ. Equ., 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.  Google Scholar

[3]

J. Bellazzini and G. Siciliano, Stable standing waves for a class of nonlinear Schrödinger-Poisson equations, Z. Angew. Math. Phys., 62 (2011), 267-280.  doi: 10.1007/s00033-010-0092-1.  Google Scholar

[4]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar

[5]

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

[6]

E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cimento, 20 (1961), 454-466.   Google Scholar

[7]

E. P. Gross, Hydrodynamics of a superfluid condensate, J. Math. Phys., 4 (1963), 195-207.  doi: 10.1063/1.1703944.  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, Concentration behavior of standing waves for almost mass critical nonlinear Schrödinger equations, J. Differential Equations, 256 (2014), 2079-2100.  doi: 10.1016/j.jde.2013.12.012.  Google Scholar

[11]

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, 30 (2016), 809-828.  doi: 10.1016/j.anihpc.2015.01.005.  Google Scholar

[12]

Q. Han and F. H. Lin, Elliptic Partial Differential Equations Courant Lect. Notes Math. vol. l, Courant Institute of Mathematical Science/AMS, New York, 2011. doi: 10.1090/cln/001.  Google Scholar

[13]

L. Jeanjean and T. J. Luo, Sharp nonexistence results of prescribed L2-norm solutions for some class of Schrödinger-Poisson and quasi-linear equations, Z. Angew. Math. Phys., 64 (2013), 937-954.  doi: 10.1007/s00033-012-0272-2.  Google Scholar

[14]

O. Kavian and F. B. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation, Michigan Math. J., 41 (1994), 151-173.  doi: 10.1307/mmj/1029004922.  Google Scholar

[15]

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

[16]

S. Li, J. L. Xiang and X. Y. Zeng, Ground states of nonlinear Choquard equations with multi-well potentials, J. Math. Phys., 57 (2016), 081515. doi: 10.1063/1.4961158.  Google Scholar

[17]

Y. Li and W. M. Ni, Radial symmetry of positive solutions of nonlinear elliptic equation in $\mathbb{R}^n$, Comm. Partial Differential Equations, 18 (1993), 1043-1054.  doi: 10.1080/03605309308820960.  Google Scholar

[18]

P. L. Lions, The concentration-compactness principle in the caclulus of variations. The locally compact case. Ⅰ, Ann. Inst H. Poincaré. Anal. Non Linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0.  Google Scholar

[19]

P. L. Lions, The concentration-compactness principle in the caclulus of variations. The locally compact case. Ⅱ, Ann. Inst H. Poincaré. Anal. Non Linéaire, 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.  Google Scholar

[20]

L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Sov. Phys. JETP, 13 (1961), 451-454.  doi: 10.11572/8649.  Google Scholar

[21]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York-London, 1978. Google Scholar

[22]

X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.  doi: 10.1007/BF02096642.  Google Scholar

[23]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolations estimates, Comm. Math. Phys., 87 (1983), 567-576.  doi: 10.1007/BF01208265.  Google Scholar

[24]

X. Y. Zeng and Y. M. Zhang, Existence and asymptotic behavior for the ground state of quasilinear elliptic equation, preprint arXiv: 1703.00183 Google Scholar

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