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

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

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 $V(x)$.

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

[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. GidasW. 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. GuoZ. Q. WangX. 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. GuoX. 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. GuoX. 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

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. GidasW. 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. GuoZ. Q. WangX. 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. GuoX. 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. GuoX. 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

[1]

Patrick Henning, Johan Wärnegård. Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation. Kinetic & Related Models, 2019, 12 (6) : 1247-1271. doi: 10.3934/krm.2019048

[2]

Norman E. Dancer. On the converse problem for the Gross-Pitaevskii equations with a large parameter. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2481-2493. doi: 10.3934/dcds.2014.34.2481

[3]

Ko-Shin Chen, Peter Sternberg. Dynamics of Ginzburg-Landau and Gross-Pitaevskii vortices on manifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1905-1931. doi: 10.3934/dcds.2014.34.1905

[4]

Thomas Chen, Nataša Pavlović. On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 715-739. doi: 10.3934/dcds.2010.27.715

[5]

Xiaoyu Zeng, Yimin Zhang. Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5263-5273. doi: 10.3934/dcds.2019214

[6]

E. Norman Dancer. On a degree associated with the Gross-Pitaevskii system with a large parameter. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1835-1839. doi: 10.3934/dcdss.2019120

[7]

Georgy L. Alfimov, Pavel P. Kizin, Dmitry A. Zezyulin. Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: Coding and method for computation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1207-1229. doi: 10.3934/dcdsb.2017059

[8]

Roy H. Goodman, Jeremy L. Marzuola, Michael I. Weinstein. Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 225-246. doi: 10.3934/dcds.2015.35.225

[9]

Yujin Guo, Xiaoyu Zeng, Huan-Song Zhou. Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3749-3786. doi: 10.3934/dcds.2017159

[10]

Jeremy L. Marzuola, Michael I. Weinstein. Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1505-1554. doi: 10.3934/dcds.2010.28.1505

[11]

Dong Deng, Ruikuan Liu. Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3175-3193. doi: 10.3934/dcdsb.2018306

[12]

Jian-Guo Liu, Jinhuan Wang. Global existence for a thin film equation with subcritical mass. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1461-1492. doi: 10.3934/dcdsb.2017070

[13]

André de Laire, Pierre Mennuni. Traveling waves for some nonlocal 1D Gross–Pitaevskii equations with nonzero conditions at infinity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 635-682. doi: 10.3934/dcds.2020026

[14]

Xiaoyu Zeng. Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1749-1762. doi: 10.3934/dcds.2017073

[15]

Rowan Killip, Satoshi Masaki, Jason Murphy, Monica Visan. The radial mass-subcritical NLS in negative order Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 553-583. doi: 10.3934/dcds.2019023

[16]

Annalisa Cesaroni, Matteo Novaga. Volume constrained minimizers of the fractional perimeter with a potential energy. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 715-727. doi: 10.3934/dcdss.2017036

[17]

Peter A. Hästö. On the existance of minimizers of the variable exponent Dirichlet energy integral. Communications on Pure & Applied Analysis, 2006, 5 (3) : 415-422. doi: 10.3934/cpaa.2006.5.415

[18]

Florian Krügel. Some properties of minimizers of a variational problem involving the total variation functional. Communications on Pure & Applied Analysis, 2015, 14 (1) : 341-360. doi: 10.3934/cpaa.2015.14.341

[19]

Annalisa Cesaroni, Serena Dipierro, Matteo Novaga, Enrico Valdinoci. Minimizers of the $ p $-oscillation functional. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6785-6799. doi: 10.3934/dcds.2019231

[20]

Jun Fan, Fusheng Lv, Lei Shi. An RKHS approach to estimate individualized treatment rules based on functional predictors. Mathematical Foundations of Computing, 2019, 2 (2) : 169-181. doi: 10.3934/mfc.2019012

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (57)
  • HTML views (119)
  • Cited by (0)

Other articles
by authors

[Back to Top]