Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations

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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 and 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.
[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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