# American Institute of Mathematical Sciences

April  2022, 21(4): 1481-1504. doi: 10.3934/cpaa.2022026

## Concentration behavior of ground states for $L^2$-critical Schrödinger Equation with a spatially decaying nonlinearity

 1 School of Mathematics and Statistics, and Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, P.O. Box 71010, Wuhan 430079, China 2 School of Mathematics and Statistics, Central China Normal University, P.O. Box 71010, Wuhan 430079, China

*Corresponding author

Received  July 2021 Revised  December 2021 Published  April 2022 Early access  February 2022

Fund Project: The first author were partially supported by the Project funded by China Postdoctoral Science Foundation No. 2019M662680 and 2019 Hubei Province Postdoctoral Science and Technology Activities Selected Funding Project

We consider ground states of the following time-independent nonlinear
 $L^2$
-critical Schrödinger equation
 $-\Delta u(x)+V(x)u(x)-a|x|^{-b}|u|^{\frac{4-2b}{N}}u(x) = \mu u(x)\,\ \hbox{in}\,\ {\mathbb{R}}^N,$
where
 $\mu\!\in\! {\mathbb{R}}$
,
 $a\!>\!0$
,
 $N\!\geq\! 1$
,
 $0\!<\!b\!<\!\min\{2,N\}$
, and
 $V(x)\!\geq\! 0$
is an external potential. We get ground states of the above equation by solving the associated constrained minimization problem. In this paper, we prove that there is a threshold
 $a^*\!>\!0$
such that minimizer exists for
 $0\!<\!a\!<\!a^*$
, and minimizer does not exist for any
 $a\!>\!a^*$
. However if
 $a\! = \!a^*$
, it is showed that whether minimizer exists depends sensitively on the value of
 $V(0)$
. Moreover if
 $V(0)\! = \!0$
, we prove that minimizers must concentrate at the origin as
 $a\nearrow a^*$
and give a detailed concentration behavior of minimizers as
 $a\nearrow a^*$
, based on which we finally prove that there is a unique minimizer when
 $a$
is close enough to
 $a^*$
.
Citation: Yong Luo, Shu Zhang. Concentration behavior of ground states for $L^2$-critical Schrödinger Equation with a spatially decaying nonlinearity. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1481-1504. doi: 10.3934/cpaa.2022026
##### References:
 [1] A. H. Ardila and V. D. Dinh, Some qualitative studies of the focusing inhomogeneous Gross-Pitaevskii equation, Z. Angew. Math. Phys., 71 (2020), 24 pp. doi: 10.1007/s00033-020-01301-z. [2] G. Baym and C. J. Pethick, Ground state properties of magnetically trapped Bose Einstein condensate rubidium gas, Phys. Rev. Lett., 76 (1996), 6-9. [3] D. Cao, S. Li and P. Luo, Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 54 (2015), 4037-4063.  doi: 10.1007/s00526-015-0930-2. [4] Y. Deng, Y. Guo and L. Lu, On the collapse and concentration of Bose-Einstein condensates with inhomogeneous attractive interactions, Calc. Var. Partial Differ. Equ., 54 (2015), 99-118.  doi: 10.1007/s00526-014-0779-9. [5] Y. Deng, Y. Guo and L. Lu, Threshold behavior and uniqueness of ground states for mass critical inhomogeneous Schrdinger equations, J. Math. Phys., 59 (2018), 011503.  doi: 10.1063/1.5008924. [6] Y. Deng, C. Lin and S. Yan, On the prescribed scalar curvature problem in ${\mathbb{R}}^N$, local uniqueness and periodicity, J. Math. Pures Appl., 104 (2015), 1013-1044.  doi: 10.1016/j.matpur.2015.07.003. [7] F. Genoud, Théorie de bifurcation et de stabilité pour une équation de Schrödinger avec une non-linéarité compacte, Ph.D thesis, EPFL, 2008. [8] F. Genoud, A uniqueness result for $\Delta u-\lambda u+V(x)u^p = 0$ on ${\mathbb{R}}^2$, Adv. Nonlinear Stud., 11 (2011), 483-491.  doi: 10.1515/ans-2011-0301. [9] F. Genoud, An Inhomogeneous, $L^{2}$-Critical, Nonlinear Schrödinger Equation, Z. Anal. Anwend., 31 (2012), 283-290.  doi: 10.4171/ZAA/1460. [10] F. Genoud and C. Stuart, Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves, Discrete Contin. Dyn. Syst., 21 (2008), 137-186.  doi: 10.3934/dcds.2008.21.137. [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1997. [12] Y. Guo, C. Lin and J. Wei, Local uniqueness and refined spike profiles of ground states for two-dimensional attractive Bose-Einstein condensates, SIAM J. Math. Anal., 49 (2017), 3671-3715.  doi: 10.1137/16M1100290. [13] Y. Guo and R. Seiringer, On the mass concentration for Bose-Einstein condensates with attactive interactions, Lett. Math. Phys., 104 (2014), 141-156.  doi: 10.1007/s11005-013-0667-9. [14] Y. Guo, Z. Wang, X. Zeng and H. 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. [15] Y. Guo, X. Zeng and H. Zhou, Energy estimates and symmetry breaking in attactive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 809-828.  doi: 10.1016/j.anihpc.2015.01.005. [16] Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lecture Note in Math., Courant Institute of Mathematical Science/AMS, New York, 2011. [17] E. H. Lieb and M. Loss, Analysis, 2$^{nd}$ edition, Graduate Studies in Mathematics, Amer. Math. Soc., Providence, 2001. doi: 10.1090/gsm/014. [18] C. Liu and V. K. Tripathi, Laser guiding in an axially nonuniform plasma channel, Phys. Plasmas, 1 (1994), 3100–3103. [19] J. F. Toland, Uniqueness of positive solutions of some semilinear Sturm-Liouville problems on the half line, Proc. Roy. Soc. Edinburgh Sect.A, 97 (1984), 259-263.  doi: 10.1017/S0308210500032042. [20] V. E. Zakharov, S. L. Musher and A. M. Rubenchik, Hamiltonian approach to the description of non-linear plasma phenomena, Phys. Rep., 129 (1985), 285-366.  doi: 10.1016/0370-1573(85)90040-7. [21] J. Zhang, Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys., 51 (2000), 498-503.  doi: 10.1007/s000330050011.

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##### References:
 [1] A. H. Ardila and V. D. Dinh, Some qualitative studies of the focusing inhomogeneous Gross-Pitaevskii equation, Z. Angew. Math. Phys., 71 (2020), 24 pp. doi: 10.1007/s00033-020-01301-z. [2] G. Baym and C. J. Pethick, Ground state properties of magnetically trapped Bose Einstein condensate rubidium gas, Phys. Rev. Lett., 76 (1996), 6-9. [3] D. Cao, S. Li and P. Luo, Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 54 (2015), 4037-4063.  doi: 10.1007/s00526-015-0930-2. [4] Y. Deng, Y. Guo and L. Lu, On the collapse and concentration of Bose-Einstein condensates with inhomogeneous attractive interactions, Calc. Var. Partial Differ. Equ., 54 (2015), 99-118.  doi: 10.1007/s00526-014-0779-9. [5] Y. Deng, Y. Guo and L. Lu, Threshold behavior and uniqueness of ground states for mass critical inhomogeneous Schrdinger equations, J. Math. Phys., 59 (2018), 011503.  doi: 10.1063/1.5008924. [6] Y. Deng, C. Lin and S. Yan, On the prescribed scalar curvature problem in ${\mathbb{R}}^N$, local uniqueness and periodicity, J. Math. Pures Appl., 104 (2015), 1013-1044.  doi: 10.1016/j.matpur.2015.07.003. [7] F. Genoud, Théorie de bifurcation et de stabilité pour une équation de Schrödinger avec une non-linéarité compacte, Ph.D thesis, EPFL, 2008. [8] F. Genoud, A uniqueness result for $\Delta u-\lambda u+V(x)u^p = 0$ on ${\mathbb{R}}^2$, Adv. Nonlinear Stud., 11 (2011), 483-491.  doi: 10.1515/ans-2011-0301. [9] F. Genoud, An Inhomogeneous, $L^{2}$-Critical, Nonlinear Schrödinger Equation, Z. Anal. Anwend., 31 (2012), 283-290.  doi: 10.4171/ZAA/1460. [10] F. Genoud and C. Stuart, Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves, Discrete Contin. Dyn. Syst., 21 (2008), 137-186.  doi: 10.3934/dcds.2008.21.137. [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1997. [12] Y. Guo, C. Lin and J. Wei, Local uniqueness and refined spike profiles of ground states for two-dimensional attractive Bose-Einstein condensates, SIAM J. Math. Anal., 49 (2017), 3671-3715.  doi: 10.1137/16M1100290. [13] Y. Guo and R. Seiringer, On the mass concentration for Bose-Einstein condensates with attactive interactions, Lett. Math. Phys., 104 (2014), 141-156.  doi: 10.1007/s11005-013-0667-9. [14] Y. Guo, Z. Wang, X. Zeng and H. 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. [15] Y. Guo, X. Zeng and H. Zhou, Energy estimates and symmetry breaking in attactive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 809-828.  doi: 10.1016/j.anihpc.2015.01.005. [16] Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lecture Note in Math., Courant Institute of Mathematical Science/AMS, New York, 2011. [17] E. H. Lieb and M. Loss, Analysis, 2$^{nd}$ edition, Graduate Studies in Mathematics, Amer. Math. Soc., Providence, 2001. doi: 10.1090/gsm/014. [18] C. Liu and V. K. Tripathi, Laser guiding in an axially nonuniform plasma channel, Phys. Plasmas, 1 (1994), 3100–3103. [19] J. F. Toland, Uniqueness of positive solutions of some semilinear Sturm-Liouville problems on the half line, Proc. Roy. Soc. Edinburgh Sect.A, 97 (1984), 259-263.  doi: 10.1017/S0308210500032042. [20] V. E. Zakharov, S. L. Musher and A. M. Rubenchik, Hamiltonian approach to the description of non-linear plasma phenomena, Phys. Rep., 129 (1985), 285-366.  doi: 10.1016/0370-1573(85)90040-7. [21] J. Zhang, Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials, Z. Angew. Math. Phys., 51 (2000), 498-503.  doi: 10.1007/s000330050011.
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