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March  2017, 37(3): 1749-1762. doi: 10.3934/dcds.2017073

Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations

Xiaoyu Zeng 1,

Department of Mathematics, School of Sciences, Wuhan University of Technology, Wuhan 430070, China

Received  May 2016 Revised  September 2016 Published  December 2016

Fund Project: The author is supported by NSFC grants 11501555 and 11471331.

We study the following minimization problem:
${d_{{a_q}}}(q): = \mathop {\inf }\limits_{\{ \int {_{{\mathbb{R}^2}}|u{|^2}dx = 1} \} } {E_{q,{a_q}}}(u),$
where the functional
$E_{q,a_q}(·)$
is given by
${{E}_{q,{{a}_{q}}}}(u):=\int_{{{\mathbb{R}}^{2}}}{(|\nabla u(x){{|}^{2}}+V(x)|u(x){{|}^{2}})}dx-\frac{2{{a}_{q}}}{q+2}\int_{{{\mathbb{R}}^{2}}}{|}u(x){{|}^{q+2}}dx.$
Here
$a_q>0, \ q∈(0,2)$
and
$V(x)$
is some type of trapping potential. Let
$a^*:= \|Q\|_2^2$
, where
$Q$
is the unique (up to translations) positive radial solution of
$Δ u-u+u^3=0$
in
$\mathbb{R}^2$
. We prove that if
$\lim_{q\nearrow2}a_q=a<a^*$
, then minimizers of
$d_{a_q}(q)$
is compact in a suitable space as
$q\nearrow2$
. On the contraty, if
$\lim_{q\nearrow2}a_q=a≥q a^*$
, by directly using asymptotic analysis, we prove that all minimizers must blow up and give the detailed asymptotic behavior of minimizers. These conclusions extend the results of Guo-Zeng-Zhou [Concentration behavior of standing waves for almost mass critical nonlinear Schrödinger equations, J. Differential Equations. 256, (2014), 2079-2100].
Keywords: Constrained variational method, energy estimates, blow-up, standing waves, nonlinear Schrödinger equation.
Mathematics Subject Classification: 35J20, 35J60.
Citation: Xiaoyu Zeng. Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1749-1762. doi: 10.3934/dcds.2017073
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics Vol. 10 Courant Institute of Mathematical Science/AMS, New York, 2003. Google Scholar

[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.  Google Scholar

[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.  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. 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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅳ. Analysis of Operators Academic Press, New York-London, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

C. A. Stuart, Bifurcation from the essential spectrum, Springer, Berlin, 45 (1983), 169-192.  doi: 10.1007/BFb0103282.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolations estimates, Comm. Math. Phys., 87 (1983), 567-576.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics Vol. 10 Courant Institute of Mathematical Science/AMS, New York, 2003. Google Scholar

[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.  Google Scholar

[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.  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. 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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅳ. Analysis of Operators Academic Press, New York-London, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

C. A. Stuart, Bifurcation from the essential spectrum, Springer, Berlin, 45 (1983), 169-192.  doi: 10.1007/BFb0103282.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolations estimates, Comm. Math. Phys., 87 (1983), 567-576.   Google Scholar

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