# American Institute of Mathematical Sciences

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. [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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##### 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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