Existence, stability and asymptotic behaviour of normalized solutions for the Davey-Stewartson system

Zhiqian He; Binhua Feng; Jiayin Liu

doi:10.3934/dcds.2022132

Article Contents

2022, Volume 42, Issue 12: 5937-5966. Doi: 10.3934/dcds.2022132

This issue Previous Article Infinitely many positive solutions for a class of semilinear elliptic equations Next Article Non simple blow ups for the Nirenberg problem on half spheres

Existence, stability and asymptotic behaviour of normalized solutions for the Davey-Stewartson system

1.
Department of Basic Teaching and Research, Qinghai University, Xining, 810016, China
2.
Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China
3.
School of Mathematics and Information Science, North Minzu University, Yinchuan, 750021, China

^*Corresponding author: Binhua Feng and Jiayin Liu
^*Corresponding author: Binhua Feng and Jiayin Liu

Received: March 2022

Revised: August 2022

Early access: September 13, 2022

Published online: September 13, 2022

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Abstract

In this paper, we undertake a comprehensive study for existence, stability and asymptotic behaviour of normalized solutions for the Davey-Stewartson system

$ -\Delta u+\omega u = a|u|^{p}u +E_1(|u|^{2})u\; \; \; in\; \mathbb{R}^2\; or \; \mathbb{R}^3,\;\;\;\;\;\;{\rm{(DS)}} $

which appears in the description of the evolution of surface water waves. In the case of $ L^2 $-critical case, i.e., $ N = 2 $, $ a>0 $ and $ 0<p<2 $, we show that normalized ground states blow up as $ c \nearrow c^*: = \|R\|^2_{L^2} $, where $ R $ is the ground state solution to equation (DS) with $ a = 0 $. We then give a detailed description for the asymptotic behavior of normalized ground states as $ c \nearrow c^* $. In the case of $ L^2 $-supercritical case, i.e., $ N = 3 $, we prove several existence and stability/instability results. We also give new criteria about global existence and blow-up for the associated evolutional equation.

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Mathematics Subject Classification: Primary: 35J50, 35Q55; Secondary: 35J20.

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References

[1]	T. Bartsch and L. Jeanjean, Normalized solutions for nonlinear Schrödinger systems, Proc. Roy. Soc. Edinburgh Sect. A, 148 (2018), 225-242. doi: 10.1017/S0308210517000087.
[2]	T. Bartsch, L. Jeanjean and N. Soave, Normalized solutions for a system of coupled cubic Schrödinger equations on $\mathbb{R}^3$, J. Math. Pures Appl., 106 (2016), 583-614. doi: 10.1016/j.matpur.2016.03.004.
[3]	T. Bartsch and N. Soave, A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, J. Funct. Anal., 272 (2017), 4998-5037. doi: 10.1016/j.jfa.2017.01.025.
[4]	T. Bartsch and N. Soave, Multiple normalized solutions for a competing system of Schrödinger equations, Calc. Var. Partial Differential Equations, 58 (2019), 24 pp. doi: 10.1007/s00526-018-1476-x.
[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 and L. Jeanjean, On dipolar quantum gases in the unstable regime, SIAM J. Math. Anal., 48 (2016), 2028-2058. doi: 10.1137/15M1015959.
[7]	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.
[8]	J. Bellazzini, L. Jeanjean and T. Luo, Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations, Proc. Lond. Math. Soc., 107 (2013), 303-339. doi: 10.1112/plms/pds072.
[9]	S. Bhattarai, On fractional Schrödinger systems of Choquard type, J. Differential Equations, 263 (2017), 3197-3229. doi: 10.1016/j.jde.2017.04.034.
[10]	D. Bonheure, J.-B. Casteras, T. Gou and L. Jeanjean, Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime, Trans. Amer. Math. Soc., 372 (2019), 2167-2212. doi: 10.1090/tran/7769.
[11]	H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490. doi: 10.1090/S0002-9939-1983-0699419-3.
[12]	T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.
[13]	T. Cazenave, An overview of the nonlinear Schrödinger equation, 2021. Available from: https://www.ljll.math.upmc.fr/cazenave/notescours.html.
[14]	T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. doi: 10.1007/BF01403504.
[15]	R. Cipolatti, On the existence of standing waves for a Davey-Stewartson system, Comm. Partial Differential Equations, 17 (1992), 967-988. doi: 10.1080/03605309208820872.
[16]	R. Cipolatti, On the instability of ground states for a Davey-Stewartson system, Ann. Inst. Henri Poincaré. Phys. Théor., 58 (1993), 85-104. http://www.numdam.org/item/?id=AIHPA_1993__58_1_85_0.
[17]	R. Cipolatti and O. Kavian, Existence of pseudo-conformally invariant solutions to the Davey-Stewartson system, J. Differential Equations, 176 (2001), 223-247. doi: 10.1006/jdeq.2000.3974.
[18]	S. Correia, Characterization of ground-states for a system of M coupled semilinear Schrödinger equations and applications, J. Differential Equations, 260 (2016), 3302-3326. doi: 10.1016/j.jde.2015.10.032.
[19]	A. Davey and K. Stewartson, On three-dimensional packets of surface waves, Proc. R. Soc. Lond. Ser. A, 338 (1974), 101-110. doi: 10.1098/rspa.1974.0076.
[20]	B. Feng and Y. Cai, Concentration for blow-up solutions of the Davey-Stewartson system in $\mathbb{R}^3$, Nonlinear Anal. Real World Appl., 26 (2015), 330-342. doi: 10.1016/j.nonrwa.2015.06.003.
[21]	B. Feng, L. Cao and J. Liu, Existence of stable standing waves for the Lee-Huang-Yang corrected dipolar Gross-Pitaevskii equation, Appl. Math. Lett., 115 (2021), 106952, 7 pp. doi: 10.1016/j.aml.2020.106952.
[22]	B. Feng, R. Chen and J. Liu, Blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation, Adv. Nonlinear Anal., 10 (2021), 311-330. doi: 10.1515/anona-2020-0127.
[23]	B. Feng, R. Chen and Q. Wang, Instability of standing waves for the nonlinear Schrödinger-Poisson equation in the $L^2$-critical case, J. Dynam. Differential Equations, 32 (2020), 1357-1370. doi: 10.1007/s10884-019-09779-6.
[24]	B. Feng, J. Ren and K. Wang, Blow-up in several points for the Davey-Stewartson system in $\mathbb{R}^2$, J. Math. Anal. Appl., 466 (2018), 1317-1326. doi: 10.1016/j.jmaa.2018.06.060.
[25]	B. Feng and Q. Wang, Strong instability of standing waves for the nonlinear Schrödinger equation in trapped dipolar quantum gases, J. Dynam. Differential Equations, 33 (2021), 1989-2008. doi: 10.1007/s10884-020-09881-0.
[26]	Z. Gan and J. Zhang, Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system, Comm. Math. Phys., 283 (2008), 93-125. doi: 10.1007/s00220-008-0456-y.
[27]	J.-M. Ghidaglia and J.-C. Saut, On the initial value problem for the Davey-Stewartson systems, Nonlinearity, 3 (1990), 475-506. doi: 10.1088/0951-7715/3/2/010.
[28]	T. Gou and L. Jeanjean, Multiple positive normalized solutions for nonlinear Schrödinger systems, Nonlinearity, 31 (2018), 2319-2345. doi: 10.1088/1361-6544/aab0bf.
[29]	T. Gou and Z. Zhang, Normalized solutions to the Chern-Simons-Schrödinger system, J. Funct. Anal., 280 (2021), 108894. doi: 10.1016/j.jfa.2020.108894.
[30]	B. Guo and B. Wang, The Cauchy problem for Davey-Stewartson systems, Comm. Pure Appl. Math., 52 (1999), 1477-1490. doi: 10.1002/(SICI)1097-0312(199912)52:12<1477::AID-CPA1>3.0.CO;2-N.
[31]	Y. Guo and R. Seiringer, On the mass concentration for Bose-Einstein condenstates with attractive interactions, Lett. Math. Phys., 104 (2014), 141-156. doi: 10.1007/s11005-013-0667-9.
[32]	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.
[33]	Y. Guo, X. Zeng and H.-S. Zhou, Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. Henri Poincaré Non Lineaire Anal., 33 (2016), 809-828. doi: 10.1016/j.anihpc.2015.01.005.
[34]	L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659. doi: 10.1016/S0362-546X(96)00021-1.
[35]	L. Jeanjean, J. Jendrej, T. Trung Le and N. Visciglia, Orbital stability of ground states for a Sobolev critical Schrödinger equation, J. Math. Pures Appl., 164 (2022), 158-179. doi: 10.1016/j.matpur.2022.06.005.
[36]	L. Jeanjean and S.-S. Lu, A mass supercritical problem revisited, Calc. Var. Partial Differential Equations, 59 (2020), No. 174, 43 pp. doi: 10.1007/s00526-020-01828-z.
[37]	L. Jeanjean and S.-S. Lu, Nonradial normalized solutions for nonlinear scalar field equations, Nonlinearity, 32 (2019), 4942-4966. doi: 10.1088/1361-6544/ab435e.
[38]	L. Jeanjean, T. Luo and Z.-Q. Wang, Multiple normalized solutions for quasi-linear Schrödinger equations, J. Differential Equations, 259 (2015), 3894-3928. doi: 10.1016/j.jde.2015.05.008.
[39]	L. Jeanjean and T. Trung Le, Multiple normalized solutions for a Sobolev critical Schrödinger equation, Math. Ann., 384 (2022), 101–134. doi: 10.1007/s00208-021-02228-0.
[40]	G. Li and X. Luo, Normalized solutions for the Chern-Simons-Schrödinger equation in $\mathbb{R}^2$, Ann. Acad. Sci. Fenn. Math., 42 (2017), 405-428. doi: 10.5186/aasfm.2017.4223.
[41]	S. Li, Y. Li and W. Yan, A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 1899-1912. doi: 10.3934/dcdss.2016077.
[42]	X. Li, J. Zhang, S. Lai and Y. Wu, The sharp threshold and limiting profile of blow-up solutions for a Davey-Stewartson system, J. Differential Equations, 250 (2011), 2197-2226. doi: 10.1016/j.jde.2010.10.022.
[43]	H. Luo and Z. Zhang, Normalized solutions to the fractional Schrödinger equations with combined nonlinearities, Calc. Var. Partial Differential Equations, 59 (2020), 35 pp. doi: 10.1007/s00526-020-01814-5.
[44]	X. Luo, Normalized standing waves for the Hartree equations, J. Differential Equations, 267 (2019), 4493-4524. doi: 10.1016/j.jde.2019.05.009.
[45]	X. Luo, J. Wei, X. Yang and M. Zhen, Normalized solutions for Schrödinger system with quadratic and cubic interactions, J. Differential Equations, 314 (2022), 56-127. doi: 10.1016/j.jde.2022.01.018.
[46]	M. Ohta, Stability of standing waves for the generalized Davey-Stewartson system, J. Dyn. Differ. Equations, 6 (1994), 325-334. doi: 10.1007/BF02218533.
[47]	M. Ohta, Blow-up solutions and strong instability of standing waves for the generalized Davey-Stewartson system in $\mathbb{R}^2$, Ann. Inst. H. Poincaré Phys. Théor., 63 (1995), 111-117. http://www.numdam.org/item/?id=AIHPA_1995__63_1_111_0.
[48]	M. Ohta, Instability of standing waves for the generalized Davey-Stewartson system, Ann. Inst. H. Poincaré Phys. Théor., 62 (1995), 69-80. http://www.numdam.org/item/?id=AIHPA_1995__62_1_69_0.
[49]	M. Ohta, Stability and instability of standing waves for the generalized Davey-Stewartson system, Differential Integral Equations, 8 (1995), 1775-1788.
[50]	T. V. Phan, Blow-up profile of Bose-Einstein condensate with singular potentials, J. Math. Phys., 58 (2017), 072301. doi: 10.1063/1.4995393.
[51]	N. Soave, Normalized ground states for the NLS equation with combined nonlinearities, J. Differential Equations, 269 (2020), 6941-6987. doi: 10.1016/j.jde.2020.05.016.
[52]	N. Soave, Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case, J. Funct. Anal., 279 (2020), 108610, 43 pp. doi: 10.1016/j.jfa.2020.108610.
[53]	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.
[54]	J. Wei and Y. Wu, Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities, J. Funct. Anal., 283 (2022), 109574, 46 pp. doi: 10.1016/j.jfa.2022.109574.
[55]	J. Zhang and S. Zhu, Sharp blow-up criteria for the Davey-Stewartson system in $\mathbb{R}^3$, Dyn. Partial Differ. Equ., 8 (2011), 239-260. doi: 10.4310/DPDE.2011.v8.n3.a4.
[56]	S. Zhu, On the Davey-Stewartson system with competing nonlinearities, J. Math. Phys., 57 (2016), 031501. doi: 10.1063/1.4942633.
[57]	S. H. Zhu, Blow-up dynamics of $L^2$ solutions for the Davey-Stewartson system, Acta Math. Sin. (Engl. Ser.), 31 (2015), 411-429. doi: 10.1007/s10114-015-4349-7.