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

Shuai Li; Jingjing Yan; Xincai Zhu

doi:10.3934/cpaa.2019005

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

References:

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[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.
[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.
[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.
[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.
[6]	E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cimento, 20 (1961), 454-466.
[7]	E. P. Gross, Hydrodynamics of a superfluid condensate, J. Math. Phys., 4 (1963), 195-207. doi: 10.1063/1.1703944.
[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, 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Sov. Phys. JETP, 13 (1961), 451-454. doi: 10.11572/8649.
[21]	M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York-London, 1978.
[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.
[23]	M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolations estimates, Comm. Math. Phys., 87 (1983), 567-576. doi: 10.1007/BF01208265.
[24]	X. Y. Zeng and Y. M. Zhang, Existence and asymptotic behavior for the ground state of quasilinear elliptic equation, preprint arXiv: 1703.00183

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.
[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.
[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.
[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.
[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.
[6]	E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cimento, 20 (1961), 454-466.
[7]	E. P. Gross, Hydrodynamics of a superfluid condensate, J. Math. Phys., 4 (1963), 195-207. doi: 10.1063/1.1703944.
[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, 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[20]	L. P. Pitaevskii, Vortex lines in an imperfect Bose gas, Sov. Phys. JETP, 13 (1961), 451-454. doi: 10.11572/8649.
[21]	M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York-London, 1978.
[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.
[23]	M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolations estimates, Comm. Math. Phys., 87 (1983), 567-576. doi: 10.1007/BF01208265.
[24]	X. Y. Zeng and Y. M. Zhang, Existence and asymptotic behavior for the ground state of quasilinear elliptic equation, preprint arXiv: 1703.00183

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American Institute of Mathematical Sciences

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

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

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