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Conserved quantities, global existence and blow-up for a generalized CH equation
Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations
Department of Mathematics, School of Sciences, Wuhan University of Technology, Wuhan 430070, China |
${d_{{a_q}}}(q): = \mathop {\inf }\limits_{\{ \int {_{{\mathbb{R}^2}}|u{|^2}dx = 1} \} } {E_{q,{a_q}}}(u),$ |
$E_{q,a_q}(·)$ |
${{E}_{q,{{a}_{q}}}}(u):=\int_{{{\mathbb{R}}^{2}}}{(|\nabla u(x){{|}^{2}}+V(x)|u(x){{|}^{2}})}dx-\frac{2{{a}_{q}}}{q+2}\int_{{{\mathbb{R}}^{2}}}{|}u(x){{|}^{q+2}}dx.$ |
$a_q>0, \ q∈(0,2)$ |
$V(x)$ |
$a^*:= \|Q\|_2^2$ |
$Q$ |
$Δ u-u+u^3=0$ |
$\mathbb{R}^2$ |
$\lim_{q\nearrow2}a_q=a<a^*$ |
$d_{a_q}(q)$ |
$q\nearrow2$ |
$\lim_{q\nearrow2}a_q=a≥q a^*$ |
References:
[1] |
W. Z. Bao and Y. Y. Cai,
Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6 (2003), 1-135.
doi: 10.3934/krm.2013.6.1. |
[2] |
T. Bartsch amd Z.-Q. Wang,
Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.
doi: 10.1080/03605309508821149. |
[3] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rat. Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[4] |
J. Byeon and Z. Q. Wang,
Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 165 (2002), 295-316.
doi: 10.1007/s00205-002-0225-6. |
[5] |
T. Cazenave,
Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics Vol. 10 Courant Institute of Mathematical Science/AMS, New York, 2003. |
[6] |
M. del Pino, M. Kowalczyk and J. C. Wei,
Concentration on curves for nonlinear schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.
doi: 10.1002/cpa.20135. |
[7] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^n$, in Mathematical analysis and applications Part A, Adv. in Math. Suppl. Stud. vol. 7, Academic Press, New York, (1981), 369–402. |
[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 for ground states of attractive Gross-Pitaevskii equations with multi-well potentials, arXiv: 1502.01839. |
[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., 2014 (256), 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. I. H. Poincaré-AN, 33 (2016), 809-828.
doi: 10.1016/j.anihpc.2015.01.005. |
[12] |
Q. Han and F. H. Lin, Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics Vol. 1 2$^{nd}$ edition, Courant Institute of Mathematical Science/AMS, New York, 2011. |
[13] |
M. K. Kwong,
Uniqueness of positive solutions of $Δ u-u+u^p=0$ in $\mathbb{R}^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[14] |
Y. Li and W.-M. Ni,
Radial symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^n$, Comm. Partial Differential Equations, 18 (1993), 1043-1054.
doi: 10.1080/03605309308820960. |
[15] |
E. H. Lieb, R. Seiringer and J. Yngvason, Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional,
Phys. Rev. A 61 (2000), 043602-1-13. |
[16] |
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.
|
[17] |
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.
|
[18] |
G. Z. Lu and J. C. Wei,
On nonlinear schrödinger equations with totally degenerate potentials, C. R. Acad. Sci. Paris., 326 (1998), 691-696.
doi: 10.1016/S0764-4442(98)80032-3. |
[19] |
M. Maeda,
On the symmetry of the ground states of nonlinear Schrödinger equation with potential, Adv. Nonlinear Stud., 10 (2010), 895-925.
doi: 10.1515/ans-2010-0409. |
[20] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅳ. Analysis of Operators Academic Press, New York-London, 1978. |
[21] |
H. A. Rose and M. I. Weinstein,
On the bound states of the nonlinear Schrödinger equation with a linear potential, Physica D, 30 (1988), 207-218.
doi: 10.1016/0167-2789(88)90107-8. |
[22] |
R. Seiringer,
Hot topics in cold gases, XVIth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, (2010), 231-245.
doi: 10.1142/9789814304634_0013. |
[23] |
C. A. Stuart,
Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math. Soc., 45 (1982), 169-192.
doi: 10.1112/plms/s3-45.1.169. |
[24] |
C. A. Stuart,
Bifurcation from the essential spectrum, Springer, Berlin, 45 (1983), 169-192.
doi: 10.1007/BFb0103282. |
[25] |
C. A. Stuart,
Bifurcation from the essential spectrum for some non-compact non-linearities, Math. Methods Applied Sci., 11 (1989), 525-542.
doi: 10.1002/mma.1670110408. |
[26] |
X. F. Wang,
On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.
doi: 10.1007/BF02096642. |
[27] |
M. I. Weinstein,
Nonlinear Schrödinger equations and sharp interpolations estimates, Comm. Math. Phys., 87 (1983), 567-576.
|
show all references
References:
[1] |
W. Z. Bao and Y. Y. Cai,
Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6 (2003), 1-135.
doi: 10.3934/krm.2013.6.1. |
[2] |
T. Bartsch amd Z.-Q. Wang,
Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.
doi: 10.1080/03605309508821149. |
[3] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rat. Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[4] |
J. Byeon and Z. Q. Wang,
Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 165 (2002), 295-316.
doi: 10.1007/s00205-002-0225-6. |
[5] |
T. Cazenave,
Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics Vol. 10 Courant Institute of Mathematical Science/AMS, New York, 2003. |
[6] |
M. del Pino, M. Kowalczyk and J. C. Wei,
Concentration on curves for nonlinear schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.
doi: 10.1002/cpa.20135. |
[7] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^n$, in Mathematical analysis and applications Part A, Adv. in Math. Suppl. Stud. vol. 7, Academic Press, New York, (1981), 369–402. |
[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 for ground states of attractive Gross-Pitaevskii equations with multi-well potentials, arXiv: 1502.01839. |
[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., 2014 (256), 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. I. H. Poincaré-AN, 33 (2016), 809-828.
doi: 10.1016/j.anihpc.2015.01.005. |
[12] |
Q. Han and F. H. Lin, Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics Vol. 1 2$^{nd}$ edition, Courant Institute of Mathematical Science/AMS, New York, 2011. |
[13] |
M. K. Kwong,
Uniqueness of positive solutions of $Δ u-u+u^p=0$ in $\mathbb{R}^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266.
doi: 10.1007/BF00251502. |
[14] |
Y. Li and W.-M. Ni,
Radial symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^n$, Comm. Partial Differential Equations, 18 (1993), 1043-1054.
doi: 10.1080/03605309308820960. |
[15] |
E. H. Lieb, R. Seiringer and J. Yngvason, Bosons in a trap: A rigorous derivation of the Gross-Pitaevskii energy functional,
Phys. Rev. A 61 (2000), 043602-1-13. |
[16] |
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.
|
[17] |
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.
|
[18] |
G. Z. Lu and J. C. Wei,
On nonlinear schrödinger equations with totally degenerate potentials, C. R. Acad. Sci. Paris., 326 (1998), 691-696.
doi: 10.1016/S0764-4442(98)80032-3. |
[19] |
M. Maeda,
On the symmetry of the ground states of nonlinear Schrödinger equation with potential, Adv. Nonlinear Stud., 10 (2010), 895-925.
doi: 10.1515/ans-2010-0409. |
[20] |
M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅳ. Analysis of Operators Academic Press, New York-London, 1978. |
[21] |
H. A. Rose and M. I. Weinstein,
On the bound states of the nonlinear Schrödinger equation with a linear potential, Physica D, 30 (1988), 207-218.
doi: 10.1016/0167-2789(88)90107-8. |
[22] |
R. Seiringer,
Hot topics in cold gases, XVIth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, (2010), 231-245.
doi: 10.1142/9789814304634_0013. |
[23] |
C. A. Stuart,
Bifurcation for Dirichlet problems without eigenvalues, Proc. London Math. Soc., 45 (1982), 169-192.
doi: 10.1112/plms/s3-45.1.169. |
[24] |
C. A. Stuart,
Bifurcation from the essential spectrum, Springer, Berlin, 45 (1983), 169-192.
doi: 10.1007/BFb0103282. |
[25] |
C. A. Stuart,
Bifurcation from the essential spectrum for some non-compact non-linearities, Math. Methods Applied Sci., 11 (1989), 525-542.
doi: 10.1002/mma.1670110408. |
[26] |
X. F. Wang,
On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244.
doi: 10.1007/BF02096642. |
[27] |
M. I. Weinstein,
Nonlinear Schrödinger equations and sharp interpolations estimates, Comm. Math. Phys., 87 (1983), 567-576.
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