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April  2022, 21(4): 1225-1247. doi: 10.3934/cpaa.2022017

On Asymptotic Properties of Semi-relativistic Hartree Equation with combined Hartree-type nonlinearities

1. 

Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

2. 

Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China

3. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

4. 

Department of Mathematics, Lanzhou University of Technology, Lanzhou, 730000, China

*Corresponding author

Received  September 2021 Revised  December 2021 Published  April 2022 Early access  December 2021

Fund Project: This work was jointly supported by the National Natural Science Foundation of China (11801519, 11971436, 12061040, 11701244), the Zhejiang Provincial Natural Science Foundation of China (LZ22A010001), the Outstanding Youth Science Fund of Gansu Province (20JR10RA111), the Natural Science Foundation of Gansu Province (21JR7RA150, 20JR5RA460)

We consider the semi-relativistic Hartree equation with combined Hartree-type nonlinearities given by
$ i\partial_t \psi = \sqrt{-\triangle+m^2}\, \psi+\beta(\frac{1}{|x|^\alpha}\ast |\psi|^2)\psi-(\frac{1}{|x|}\ast |\psi|^2)\psi\ \ \ \text{on $\mathbb{R}^3$.} $
where
$ 0<\alpha<1 $
and
$ \beta>0 $
. Firstly we study the existence and stability of the maximal ground state
$ \psi_\beta $
at
$ N = N_c $
, where
$ N_c $
is a threshold value and can be regarded as "Chandrasekhar limiting mass". Secondly, we analyse blow-up behaviours of maximal ground states
$ \psi_\beta $
when
$ \beta\rightarrow 0^+ $
, and the optimal blow-up rate with respect to
$ \beta $
will be calculated.
Citation: Qingxuan Wang, Binhua Feng, Yuan Li, Qihong Shi. On Asymptotic Properties of Semi-relativistic Hartree Equation with combined Hartree-type nonlinearities. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1225-1247. doi: 10.3934/cpaa.2022017
References:
[1]

S. Cingolani and S. Secchi, Ground states for the pseudo-relativistic Hartree equation with external potential, Proc. Roy. Soc. Edinb. A, 145 (2015), 73-90.  doi: 10.1017/S0308210513000450.

[2]

Y. Cho and T. Ozawa, On the semi-relativistic Hartree type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074.  doi: 10.1137/060653688.

[3]

Y. DengY. 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.

[4]

A. Elgart and B. Schlein, Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60 (2007), 500-545.  doi: 10.1002/cpa.20134.

[5]

B. Feng, On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities, J. Evol. Equ., 18 (2018), 203-220.  doi: 10.1007/s00028-017-0397-z.

[6]

R. C. FetecauY. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.  doi: 10.1088/0951-7715/24/10/002.

[7]

J. FröhlichB. LarsG. Jonsson and E. Lenzmann, Boson stars as solitary waves, Commun. Math. Phys., 274 (2007), 1-30.  doi: 10.1007/s00220-007-0272-9.

[8]

J. Fröhlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math., 60 (2007), 1691-1705.  doi: 10.1002/cpa.20186.

[9]

Y. Guo and R. Seiringer, On the Mass concentration for Bose-Einstein condensation with attractive interactions, Lett. Math. Phys., 104 (2014), 141-156.  doi: 10.1007/s11005-013-0667-9.

[10]

Y. Guo and X. Zeng, Ground states of pseudo-relativistic boson stars under the critical stellar mass, Ann. I. H. Poincaré, 34 (2017), 1611-1632.  doi: 10.1016/j.anihpc.2017.04.001.

[11]

Y. GuoX. Zeng and H. Zhou, Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Pioncaré, 33 (2016), 809-828.  doi: 10.1016/j.anihpc.2015.01.005.

[12]

S. Herr and E. Lenzmann, The Boson star equation with initial data of low regularity, Nonlinear Anal., 97 (2014), 125-137.  doi: 10.1016/j.na.2013.11.023.

[13]

D. Holm and V. Putkaradze, Formation of clumps and patches in selfaggregation of finite-size particles, Phys. D, 220 (2006), 183-196.  doi: 10.1016/j.physd.2006.07.010.

[14]

E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64.  doi: 10.1007/s11040-007-9020-9.

[15]

E. Lenzmann, Uniqueness of ground states for pseudo-relativistic Hartree equations, Anal. Partial Differ. Equ., 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.

[16]

E. Lenzmann and M. Lewin, On singularity formation for the $L^2$-critical Boson star equation, Nonlinearity, 24 (2011), 3515-3540.  doi: 10.1088/0951-7715/24/12/009.

[17]

E. H. Lieb and H. T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics., Commun. Math. Phys., 112 (1987), 147-174. 

[18]

E. H. Lieb and M. Loss, Analysis 2ed. Grad. Stud. Math., Amer. Math. Soc., 2001. doi: 10.1090/gsm/014.

[19]

X. Luo, Normalized standing waves for the Hartree equations, J. Differ. Equ., 267 (2019), 4493-4524.  doi: 10.1016/j.jde.2019.05.009.

[20]

A. Michelangeli and B. Schlein, Dynamical collapse of boson stars, Commun. Math. Phys., 311 (2012), 645-687.  doi: 10.1007/s00220-011-1341-7.

[21]

D. T. Nguyen, On Blow-up Profile of Ground States of Boson Stars with External Potential, J. Stat. Phys., 169 (2017), 395-422.  doi: 10.1007/s10955-017-1872-1.

[22]

F. Pusateri, Modified Scattering for the Boson Star Equation, Commun. Math. Phys., 332 (2014), 1203-1234.  doi: 10.1007/s00220-014-2094-x.

[23]

Q. Shi and C. Peng, Well-posedness for semirelativistic Schrödinger equation with power-type nonlinearity, Nonl. Anal., 178 (2019), 133-144.  doi: 10.1016/j.na.2018.07.012.

[24]

N. Soave, Normalized ground states for the NLS equation with combined nonlinearities, J. Differ. Equ., 269 (2020), 6941-6987.  doi: 10.1016/j.jde.2020.05.016.

[25]

N. Soave, Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case, J. Funct. Anal., 279 (2020), 1-43.  doi: 10.1016/j.jfa.2020.108610.

[26] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. 
[27]

T. TaoM. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Commun. Partial Differ. Equ., 32 (2007), 1281-1343.  doi: 10.1080/03605300701588805.

[28]

C. TopazA. Bertozzi and M. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.

[29]

Q. Wang and D. Zhao, Existence and mass concentration of 2D attractive Bose-Einstein condensates with periodic potentials, J. Differ. Equ., 262 (2017), 2684-2704.  doi: 10.1016/j.jde.2016.11.004.

[30]

J. Yang and J. Yang, Existence and mass concentration of pseudo-relativistic Hartree equation, J. Math. Phys., 58 (2017), 1-22.  doi: 10.1063/1.4996576.

[31]

V. C. Zelati and M. Nolasco, Ground states for pseudo-relativistic Hartree equations of critical type, Rev. Mat. Ibero., 29 (2013), 1421-1436.  doi: 10.4171/RMI/763.

[32]

X. Zeng and L. Zhang, Normalized solutions for Schrödinger-Poisson-Slater equations with unbounded potentials, J. Math. Anal. Appl., 452 (2017), 47-61.  doi: 10.1016/j.jmaa.2017.02.053.

show all references

References:
[1]

S. Cingolani and S. Secchi, Ground states for the pseudo-relativistic Hartree equation with external potential, Proc. Roy. Soc. Edinb. A, 145 (2015), 73-90.  doi: 10.1017/S0308210513000450.

[2]

Y. Cho and T. Ozawa, On the semi-relativistic Hartree type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074.  doi: 10.1137/060653688.

[3]

Y. DengY. 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.

[4]

A. Elgart and B. Schlein, Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60 (2007), 500-545.  doi: 10.1002/cpa.20134.

[5]

B. Feng, On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities, J. Evol. Equ., 18 (2018), 203-220.  doi: 10.1007/s00028-017-0397-z.

[6]

R. C. FetecauY. Huang and T. Kolokolnikov, Swarm dynamics and equilibria for a nonlocal aggregation model, Nonlinearity, 24 (2011), 2681-2716.  doi: 10.1088/0951-7715/24/10/002.

[7]

J. FröhlichB. LarsG. Jonsson and E. Lenzmann, Boson stars as solitary waves, Commun. Math. Phys., 274 (2007), 1-30.  doi: 10.1007/s00220-007-0272-9.

[8]

J. Fröhlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars, Comm. Pure Appl. Math., 60 (2007), 1691-1705.  doi: 10.1002/cpa.20186.

[9]

Y. Guo and R. Seiringer, On the Mass concentration for Bose-Einstein condensation with attractive interactions, Lett. Math. Phys., 104 (2014), 141-156.  doi: 10.1007/s11005-013-0667-9.

[10]

Y. Guo and X. Zeng, Ground states of pseudo-relativistic boson stars under the critical stellar mass, Ann. I. H. Poincaré, 34 (2017), 1611-1632.  doi: 10.1016/j.anihpc.2017.04.001.

[11]

Y. GuoX. Zeng and H. Zhou, Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Pioncaré, 33 (2016), 809-828.  doi: 10.1016/j.anihpc.2015.01.005.

[12]

S. Herr and E. Lenzmann, The Boson star equation with initial data of low regularity, Nonlinear Anal., 97 (2014), 125-137.  doi: 10.1016/j.na.2013.11.023.

[13]

D. Holm and V. Putkaradze, Formation of clumps and patches in selfaggregation of finite-size particles, Phys. D, 220 (2006), 183-196.  doi: 10.1016/j.physd.2006.07.010.

[14]

E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64.  doi: 10.1007/s11040-007-9020-9.

[15]

E. Lenzmann, Uniqueness of ground states for pseudo-relativistic Hartree equations, Anal. Partial Differ. Equ., 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.

[16]

E. Lenzmann and M. Lewin, On singularity formation for the $L^2$-critical Boson star equation, Nonlinearity, 24 (2011), 3515-3540.  doi: 10.1088/0951-7715/24/12/009.

[17]

E. H. Lieb and H. T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics., Commun. Math. Phys., 112 (1987), 147-174. 

[18]

E. H. Lieb and M. Loss, Analysis 2ed. Grad. Stud. Math., Amer. Math. Soc., 2001. doi: 10.1090/gsm/014.

[19]

X. Luo, Normalized standing waves for the Hartree equations, J. Differ. Equ., 267 (2019), 4493-4524.  doi: 10.1016/j.jde.2019.05.009.

[20]

A. Michelangeli and B. Schlein, Dynamical collapse of boson stars, Commun. Math. Phys., 311 (2012), 645-687.  doi: 10.1007/s00220-011-1341-7.

[21]

D. T. Nguyen, On Blow-up Profile of Ground States of Boson Stars with External Potential, J. Stat. Phys., 169 (2017), 395-422.  doi: 10.1007/s10955-017-1872-1.

[22]

F. Pusateri, Modified Scattering for the Boson Star Equation, Commun. Math. Phys., 332 (2014), 1203-1234.  doi: 10.1007/s00220-014-2094-x.

[23]

Q. Shi and C. Peng, Well-posedness for semirelativistic Schrödinger equation with power-type nonlinearity, Nonl. Anal., 178 (2019), 133-144.  doi: 10.1016/j.na.2018.07.012.

[24]

N. Soave, Normalized ground states for the NLS equation with combined nonlinearities, J. Differ. Equ., 269 (2020), 6941-6987.  doi: 10.1016/j.jde.2020.05.016.

[25]

N. Soave, Normalized ground states for the NLS equation with combined nonlinearities: the Sobolev critical case, J. Funct. Anal., 279 (2020), 1-43.  doi: 10.1016/j.jfa.2020.108610.

[26] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. 
[27]

T. TaoM. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Commun. Partial Differ. Equ., 32 (2007), 1281-1343.  doi: 10.1080/03605300701588805.

[28]

C. TopazA. Bertozzi and M. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.

[29]

Q. Wang and D. Zhao, Existence and mass concentration of 2D attractive Bose-Einstein condensates with periodic potentials, J. Differ. Equ., 262 (2017), 2684-2704.  doi: 10.1016/j.jde.2016.11.004.

[30]

J. Yang and J. Yang, Existence and mass concentration of pseudo-relativistic Hartree equation, J. Math. Phys., 58 (2017), 1-22.  doi: 10.1063/1.4996576.

[31]

V. C. Zelati and M. Nolasco, Ground states for pseudo-relativistic Hartree equations of critical type, Rev. Mat. Ibero., 29 (2013), 1421-1436.  doi: 10.4171/RMI/763.

[32]

X. Zeng and L. Zhang, Normalized solutions for Schrödinger-Poisson-Slater equations with unbounded potentials, J. Math. Anal. Appl., 452 (2017), 47-61.  doi: 10.1016/j.jmaa.2017.02.053.

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