# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2022017
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## 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 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 & Applied Analysis, doi: 10.3934/cpaa.2022017
##### References:
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Equ., 32 (2007), 1281-1343.  doi: 10.1080/03605300701588805.  Google Scholar [28] C. Topaz, A. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [3] 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] R. C. Fetecau, Y. 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.  Google Scholar [7] J. Fröhlich, B. Lars, G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Commun. Math. Phys., 274 (2007), 1-30.  doi: 10.1007/s00220-007-0272-9.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] Y. Guo, X. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [18] E. H. Lieb and M. Loss, Analysis 2ed. Grad. Stud. Math., Amer. Math. Soc., 2001. doi: 10.1090/gsm/014.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] F. Pusateri, Modified Scattering for the Boson Star Equation, Commun. Math. Phys., 332 (2014), 1203-1234.  doi: 10.1007/s00220-014-2094-x.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.   Google Scholar [27] T. Tao, M. 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.  Google Scholar [28] C. Topaz, A. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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