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Multiplicity of solutions for a nonhomogeneous quasilinear elliptic problem with critical growth
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A positive solution for an asymptotically cubic quasilinear Schrödinger equation
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 |
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 $V(x)$.
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 |
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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