Blow-up profile of normalized solutions for fractional nonlinear Schrödinger equation with negative potentials

Zaizheng Li; Haijun Luo; Zhitao Zhang

doi:10.3934/dcds.2024089

Article Contents

2025, Volume 45, Issue 1: 160-188. Doi: 10.3934/dcds.2024089

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Blow-up profile of normalized solutions for fractional nonlinear Schrödinger equation with negative potentials

1.
School of Mathematical Sciences, Hebei Center for Applied Mathematics, Hebei Normal University, Shijiazhuang 050024, Hebei, China
2.
School of Mathematics, Hunan Provincial Key Laboratory of Intelligent Information Processing and Applied Mathematics, Hunan University, Changsha 410082, Hunan, China
3.
HLM, Academy of Mathematics and Systems Science, the Chinese Academy of Sciences, Beijing 100190, China
4.
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
5.
School of Mathematical Science, Jiangsu University, Zhenjiang 212013, Jiangsu, China

^*Corresponding author: Haijun Luo
^*Corresponding author: Haijun Luo

Received: May 2024

Revised: June 2024

Early access: July 2024

Published: January 2025

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Abstract

We investigate the existence and blow-up profile of normalized solutions to the fractional nonlinear Schrödinger equation

$ \begin{align} \begin{cases} (-\Delta)^su+V(x)u+\lambda u = |u|^{\frac{4s}{N}}u, \ \text{in}\ \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2\mathrm{d}x = \alpha, \end{cases} \end{align} $

with $ N\ge 2, s\in(0, 1), \alpha>0 $, $ \lambda\in\mathbb{R} $, and negative potentials $ V(x) $. Firstly, we prove the existence and nonexistence of normalized solutions for negative potentials $ V\in L^p(\mathbb{R}^N)+L^{q}(\mathbb{R}^N) $ with $ \frac{N}{2s}\le p<q < \infty $ in the mass-critical setting. Secondly, we refine the nonexistence results and analyze the asymptotic behavior of the minimizers $ \{u_{\alpha}\} $ as $ \alpha \nearrow \alpha^* $ for two types of potentials: a bounded potential $ V(x) = -\frac{\eta}{(1+|x|^{2})^s} $, and a singular potential $ V(x) = -\gamma_1|x-a_1|^{-m_1}-\gamma_2|x-a_2|^{-m_2} $. Our study provides precise energy estimates and sharp blow-up rates in the convergence of minimizers $ \{u_{\alpha}\} $.

Keywords:

Fractional nonlinear Schrödinger equation,
standing wave,
normalized solution,
blow-up profile,
mass critical nonlinearity.

Mathematics Subject Classification: Primary: 35R11, 35B40; Secondary: 35B44.

Citation:

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Table 1. Existence and nonexistence of global minimizers

	$ L^p(\mathbb{R}^N)+L^q(\mathbb{R}^N) $	$ -\frac{\eta}{(1+\|x\|^2)^s} $	$ -\gamma_1\|x-a_1\|^{-m_1}-\gamma_2\|x-a_2\|^{-m_2} $
$ \alpha_* $	$ \alpha_* $ is unknown	$ \alpha_>0 $ for $ \eta $ small; $ \alpha_=0 $ for $ \eta $ large	$ \alpha_*=0 $
$ (0,\alpha_*] $	$ E(\alpha) $ is unknown	$ E(\alpha)= 0 $; no global minimizer	$ \emptyset $
$ (\alpha_,\alpha^) $	$ E(\alpha)<0 $; global minimizer exists
$ \alpha^* $	no global minimizer
$ \alpha^* $	$ E(\alpha^)=\frac{\text{ess inf} V}{2}\alpha^ $	$ E(\alpha^)=-\frac{\eta}{2}\alpha^ $	$ E(\alpha^*)=-\infty $
$ (\alpha^*,+\infty) $	$ E(\alpha)=-\infty $; no global minimizer

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References

[1]	D. Applebaum, Lévy Processes and Stochastic Calculus, 2$^{nd}$ edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.
[2]	T. Bartsch, H. Li and W. Zou, Existence and asymptotic behavior of normalized ground states for Sobolev critical Schrödinger systems, Calc. Var. Partial Differential Equations, 62 (2023), Paper No. 9, 34 pp. doi: 10.1007/s00526-022-02355-9.
[3]	T. Bartsch, R. Molle, M. Rizzi and G. Verzini, Normalized solutions of mass supercritical Schrödinger equations with potential, Comm. Partial Differential Equations, 46 (2021), 1729-1756. doi: 10.1080/03605302.2021.1893747.
[4]	T. Bartsch, S. Qi and W. Zou, Normalized solutions to Schrödinger equations with potential and inhomogeneous nonlinearities on large smooth domains, Math. Ann., (2024). doi: 10.1007/s00208-024-02857-1.
[5]	T. Bartsch, X. Zhong and W. Zou, Normalized solutions for a coupled Schrödinger system, Math. Ann., 380 (2021), 1713-1740. doi: 10.1007/s00208-020-02000-w.
[6]	J. Bellazzini, N. Boussaïd, L. Jeanjean and N. Visciglia, Existence and stability of standing waves for supercritical NLS with a partial confinement, Comm. Math. Phys., 353 (2017), 229-251. doi: 10.1007/s00220-017-2866-1.
[7]	A. Burchard, A Short Course on Rearrangement Inequalities, Lecture Notes, IMDEA Winter School, Madrid, 2019. Available from: https://www.math.utoronto.ca/almut/rearrange.pdf.
[8]	M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phys., 53 (2012), 043507, 7 pp. doi: 10.1063/1.3701574.
[9]	Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa, On the orbital stability of fractional Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 1267-1282. doi: 10.3934/cpaa.2014.13.1267.
[10]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[11]	Y. Ding and X. Zhong, Normalized solution to the Schrödinger equation with potential and general nonlinear term: Mass super-critical case, J. Differential Equations, 334 (2022), 194-215. doi: 10.1016/j.jde.2022.06.013.
[12]	V. D. Dinh, Existence, non-existence and blow-up behaviour of minimizers for the mass-critical fractional non-linear Schrödinger equations with periodic potentials, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 3252-3292. doi: 10.1017/prm.2019.64.
[13]	M. Du, L. Tian, J. Wang and F. Zhang, Existence of normalized solutions for nonlinear fractional Schrödinger equations with trapping potentials, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 617-653. doi: 10.1017/prm.2018.41.
[14]	B. Dyda, A fractional order Hardy inequality, Illinois J. Math., 48 (2024), 575-588. doi: 10.1215/ijm/1258138400.
[15]	R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb {R}$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9.
[16]	R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726. doi: 10.1002/cpa.21591.
[17]	H. Guo and H.-S. Zhou, A constrained variational problem arising in attractive Bose-Einstein condensate with ellipse-shaped potential, Appl. Math. Lett., 87 (2019), 35-41. doi: 10.1016/j.aml.2018.07.023.
[18]	Y. 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.
[19]	Y. Guo, Z.-Q. Wang, X. 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.
[20]	Y. Guo, X. Zeng and H.-S. Zhou, Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 33 (2016), 809-828. doi: 10.1016/j.anihpc.2015.01.005.
[21]	Q. He and W. Long, The concentration of solutions to a fractional Schrödinger equation, Z. Angew. Math. Phys., 67 (2016), Art. 9, 19 pp. doi: 10.1007/s00033-015-0607-x.
[22]	N. Ikoma and Y. Miyamoto, Stable standing waves of nonlinear Schrödinger equations with potentials and general nonlinearities, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 48, 20 pp. doi: 10.1007/s00526-020-1703-0.
[23]	K. Kirkpatrick, E. Lenzmann and G. Staffilani, On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317 (2013), 563-591. doi: 10.1007/s00220-012-1621-x.
[24]	N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305. doi: 10.1016/S0375-9601(00)00201-2.
[25]	S. Li, J. Yan and X. Zhu, Constraint minimizers of perturbed Gross-Pitaevskii energy functionals in $\mathbb{R}^N$, Commun. Pure Appl. Anal., 18 (2019), 65-81. doi: 10.3934/cpaa.2019005.
[26]	S. Li and X. Zhu, Mass concentration and local uniqueness of ground states for $L^2$-subcritical nonlinear Schrödinger equations, Z. Angew. Math. Phys., 70 (2019), Paper No. 34, 26 pp. doi: 10.1007/s00033-019-1077-3.
[27]	Z. Li, Q. Zhang and Z. Zhang, Standing waves of fractional Schrödinger equations with potentials and general nonlinearities, Anal. Theory Appl., 39 (2023), 357-377. doi: 10.4208/ata.OA-2022-0012.
[28]	M.-Q. Liu and W. Zou, Normalized solutions to fractional Schrödinger equation with potentials, Discrete Contin. Dyn. Syst. Ser. S, 16 (2023), 3194-3211. doi: 10.3934/dcdss.2023188.
[29]	H. Luo and Z. Zhang, Normalized solutions to the fractional Schrödinger equations with combined nonlinearities, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 143, 35 pp. doi: 10.1007/s00526-020-01814-5.
[30]	G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316282397.
[31]	R. Molle, G. Riey and G. Verzini, Normalized solutions to mass supercritical Schrödinger equations with negative potential, J. Differential Equations, 333 (2022), 302-331. doi: 10.1016/j.jde.2022.06.012.
[32]	S. Peng and A. Xia, Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential, Commun. Pure Appl. Anal., 20 (2021), 3723-3744. doi: 10.3934/cpaa.2021128.
[33]	T. V. Phan, Blow-up profile of Bose-Einstein condensate with singular potentials, J. Math. Phys., 58 (2017), 072301, 10 pp. doi: 10.1063/1.4995393.
[34]	P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, No. 65. American Mathematical Soc., 1986. doi: 10.1090/cbms/065.
[35]	Q. Wang and D. Zhao, Existence and mass concentration of 2D attractive Bose-Einstein condensates with periodic potentials, J. Differential Equations, 262 (2017), 2684-2704. doi: 10.1016/j.jde.2016.11.004.
[36]	Y. Wang, X. Y. Zeng and H. S. Zhou, Asymptotic behavior of least energy solutions for a fractional Laplacian eigenvalue problem on $\mathbb{R}^N$, Acta Math. Sin. (Engl. Ser.), 39 (2023), 707-727. doi: 10.1007/s10114-023-1074-5.
[37]	X. Zhang, M. Squassina and J. Zhang, Multiplicity of normalized solutions for the fractional Schrödinger equation with potentials, Mathematics., 12 (2024), 772. doi: 10.3390/math12050772.
[38]	Z. Zhang, Variational, Topological, and Partial Order Methods with their Applications, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-30709-6.
[39]	J. Zuo, C. Liu and C. Vetro, Normalized Solutions to the Fractional Schrödinger Equation with Potential, Mediterr. J. Math., 20 (2023), Paper No. 216, 12 pp. doi: 10.1007/s00009-023-02422-1.