November  2021, 41(11): 5209-5241. doi: 10.3934/dcds.2021074

Classification of solutions to a nonlocal equation with doubly Hardy-Littlewood-Sobolev critical exponents

1. 

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

2. 

Department of Mathematics, Yunnan Normal University, Kunming 650500, China

* Corresponding author: Shunneng Zhao

Received  October 2020 Revised  March 2021 Published  November 2021 Early access  April 2021

Fund Project: The first author is supported by NSFC(11571317, 11971436) and ZJNSF(LD19A010001), the second author is supported by NSFC(11771385)

We consider the following nonlocal critical equation
$\begin{equation} -\Delta u = (I_{\mu_1}\ast|u|^{2_{\mu_1}^\ast})|u|^{2_{\mu_1}^\ast-2}u +(I_{\mu_2}\ast|u|^{2_{\mu_2}^\ast})|u|^{2_{\mu_2}^\ast-2}u,\; x\in\mathbb{R}^N, \;\;\;\;\;\;\;(1) \end{equation}$
where
$ 0<\mu_1,\mu_2<N $
if
$ N = 3 $
or
$ 4 $
, and
$ N-4\leq\mu_1,\mu_2<N $
if
$ N\geq5 $
,
$ 2_{\mu_{i}}^\ast: = \frac{N+\mu_i}{N-2}(i = 1,2) $
is the upper critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and
$ I_{\mu_i} $
is the Riesz potential
$ \begin{equation*} I_{\mu_i}(x) = \frac{\Gamma(\frac{N-\mu_i}{2})}{\Gamma(\frac{\mu_i}{2})\pi^{\frac{N}{2}}2^{\mu_i}|x|^{N-\mu_i}}, \; i = 1,2, \end{equation*} $
with
$ \Gamma(s) = \int_{0}^{\infty}x^{s-1}e^{-x}dx $
,
$ s>0 $
. Firstly, we prove the existence of the solutions of the equation (1). We also establish integrability and
$ C^\infty $
-regularity of solutions and obtain the explicit forms of positive solutions via the method of moving spheres in integral forms. Finally, we show that the nondegeneracy of the linearized equation of (1) at
$ U_0,V_0 $
when
$ \max\{\mu_1,\mu_2\}\rightarrow0 $
and
$ \min\{\mu_1,\mu_2\}\rightarrow N $
, respectively.
Citation: Minbo Yang, Fukun Zhao, Shunneng Zhao. Classification of solutions to a nonlocal equation with doubly Hardy-Littlewood-Sobolev critical exponents. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5209-5241. doi: 10.3934/dcds.2021074
References:
[1]

A. Alexandrov, Uniqueness theorems for surfaces in large V, Am.of Math. Soc.Transl, 21 (1962), 412-416. 

[2]

T. Aubin, Best constans in the Sobolev imbedding theorem: the Yamabe problem, Ann.of Math. Stud., 115 (1989), 173-184. 

[3]

T. BartchT. Weth and M. Willem, A sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator, Calc. Var. Partial Differ. Equ., 18 (2003), 253-268.  doi: 10.1007/s00526-003-0198-9.

[4]

G. Bianchiand and H. Egnell, A note on the sobolev inequality, J. Funct. Anal., 100 (1991), 18-24.  doi: 10.1016/0022-1236(91)90099-Q.

[5]

L. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior semilinear elliptic equations with critical Sobolev groth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.

[6]

W. ChenW. JinC. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalitities and systems of integral equations, Discrete Contin. Dyn. Syst. Suppl., 14 (2005), 164-172. 

[7]

C. Chen and C. Lin, Uniqueness of the ground state solution of $\Delta u+f(u)$ in $\mathbb{R}^n$, $n\geq3$, Commun. Partial Diff. Equ., 16 (1991), 1549-1572.  doi: 10.1080/03605309108820811.

[8]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.

[9]

W. Chen and C. Li, On Nirenberg and related problemsa necessary and sufficient condition, Commun. Pure Appl. Math., 48 (1995), 657-667.  doi: 10.1002/cpa.3160480606.

[10]

L. ChenZ. Liu and G. Lu, Symmetry and Regularity of solutions to the Weighted Hardy-Sobolev Type System, Adv. Nonlinear Stud., 16 (2016), 1-13.  doi: 10.1515/ans-2015-5005.

[11]

W. ChenC. Li and B. Ou, Classification of solutions for a systen of integral equations, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[12]

C. Coffman, On the positive solutions of boundary value problems for a class of nonlinear differential equatins, J. Diff. Eq., 3 (1967), 92-111.  doi: 10.1016/0022-0396(67)90009-5.

[13]

W. Dai, J. Huang, Y. Qin, B. Wang and Y. Fang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete Contin. Dyn. Syst., 39 (2019), 1389C–1403. doi: 10.3934/dcds.2018117.

[14]

Y. DingF. Gao and M. Yang, Semiclassical states for Choquard type equations with critical growth: Critical frequency case, Nonlinearity, 33 (2020), 6695-6728.  doi: 10.1088/1361-6544/aba88d.

[15]

L. Du, F. Gao and M. Yang, Existence and qualitative analysis for nonlinear weighted Choqaurd equations, preprint, arXiv: 1810.11759.

[16]

L. Du and M. Yang, Uniqueness and nondegeneracy of solutions for a critical nonlocal equation, Discrete Contin. Dyn. Syst., 39 (2019), 5847-5866.  doi: 10.3934/dcds.2019219.

[17]

F. GaoE. SilvaM. Yang and J. Zhou, Existence of solutions for critical Choquard equations via the concentration compactness method, Proc. Roy. Soc. Edinb. A, 150 (2020), 921-954.  doi: 10.1017/prm.2018.131.

[18]

F. Gao, M. Yang and J. Zhou, Existence of multiple semiclassical solutions for a critical Choquard equation with indefinite potential, Nonlinear Anal. TMA, 195 (2020), 111817. doi: 10.1016/j.na.2020.111817.

[19]

J. Giacomoni, Y. Wei and M. Yang, Nondegeneracy of solutions for a critical Hartree equation, Nonlinear Anal. TMA, 2020, 111969. doi: 10.1016/j.na.2020.111969.

[20]

B. GidasW. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^N$, Adv. Math. Sppl. Stud., A (1981), 369-402. 

[21]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.

[22]

L. Guo, T. Hu, S. Peng and W. Shuai, Existence and uniqueness of solutions for Choquard equation involving Hard-Littlewood-Sobolev critical exponent, Calc. Var. partial Diff. Equ., 58 (2019), 128, 34 pp. doi: 10.1007/s00526-019-1585-1.

[23]

C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457.  doi: 10.1007/s00526-006-0013-5.

[24]

H. Kaper and M. Kwong., Uniqueness of non-negative solutions of a class of semi-linear elliptic equations, in Nonlinear Diffusion Equations and Their Equilibrium States, Springer-Verlag, New York, 1988, 1-17. doi: 10.1007/978-1-4612-0873-0.

[25]

M. Kwong, Uniqueness of positive solutions of $\Delta u+u^p = 0\mathbb{R}^n$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.

[26]

Y. Lei, Qualitative analysis for the Hartree-type equations, SIAM. J. Math. Anal., 45 (2013), 388-406.  doi: 10.1137/120879282.

[27]

C. Li, Local asymptotic symmetry of singular solutions of to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/s002220050023.

[28]

Y. Li, Remark on some confomally invariant integral equations: The method of moving spheres, J. Eur.Math.Soc., 2 (2004), 153-180. 

[29]

Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8.

[30]

E. H. Lieb, Existence and uniquenss of the minimizing solution of choquard's nonlinear equation, Stud. Appl. Math., 57 (1976/1977), 93-105.  doi: 10.1002/sapm197757293.

[31]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.

[32]

E. H. Lieb and M. Loss, Graduate Studies in Mathematics, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1007/978-1-4612-0873-0.

[33]

P. L Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.

[34]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.

[35]

C. MaW. Chen and C. Li, Rugularity of solutions for an integral system of Wolff type, Advances in Mathematics, 226 (2011), 2676-2699.  doi: 10.1016/j.aim.2010.07.020.

[36]

K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u) = 0$ in $\mathbb{R}^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145.  doi: 10.1007/BF00275874.

[37]

I. M. MorozR. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.

[38]

V. Moroz and J. Van Schaftingen, Ground states of nonlinear choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.

[39]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12. doi: 10.1142/S0219199715500054.

[40]

W. Ni and R. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $\Delta u+f(u, r) = 0$, Comm. Pure and Appl. Math., 38 (1985), 69-108.  doi: 10.1002/cpa.3160380105.

[41]

P. Padilla, On Some Nonlinear Elliptic Equations, Ph.D Thesis, Courant Institute, 1994.

[42]

S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, (1954). doi: 10.1007/978-1-4612-0873-0.

[43]

L. Peletier and J. Serrin, Uniqueness of solutions of semilinear equations in $\mathbb{R}^n$, J. Diff. Eq., 61 (1986), 380-397.  doi: 10.1016/0022-0396(86)90112-9.

[44]

J. Seok, Limit profiles and uniqueness of ground states to the nonliner Choquard equations, Advances in Nonlinear Analysis, 20 (2019), 207-228.  doi: 10.1515/anona-2017-0182.

[45]

J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923.  doi: 10.1512/iumj.2000.49.1893.

[46]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura. Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.

[47]

P. Tod and I. M. Moroz, An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12 (1999), 201-216.  doi: 10.1088/0951-7715/12/2/002.

[48]

J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905. doi: 10.1063/1.3060169.

[49]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.  doi: 10.1007/s002080050258.

[50]

M. Yang and X. Zhou, On a Coupled Schödinger System with Stein-Weiss Type Convolution Part, J. Geom. Anal., (2021). doi: 10.1007/s12220-021-00645-w.

[51]

L. Zhang and C. Lin, Uniqueness of ground state solutions, Acta Math. Sci., 8 (1988), 449-468.  doi: 10.1016/S0252-9602(18)30321-7.

[52]

Y. ZhenF. GaoZ. Shen and M. Yang, On a class of coupled critical Hartree system with deepening potential, Math. Meth. Appl. Sci., 44 (2021), 772-798.  doi: 10.1002/mma.6785.

show all references

References:
[1]

A. Alexandrov, Uniqueness theorems for surfaces in large V, Am.of Math. Soc.Transl, 21 (1962), 412-416. 

[2]

T. Aubin, Best constans in the Sobolev imbedding theorem: the Yamabe problem, Ann.of Math. Stud., 115 (1989), 173-184. 

[3]

T. BartchT. Weth and M. Willem, A sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator, Calc. Var. Partial Differ. Equ., 18 (2003), 253-268.  doi: 10.1007/s00526-003-0198-9.

[4]

G. Bianchiand and H. Egnell, A note on the sobolev inequality, J. Funct. Anal., 100 (1991), 18-24.  doi: 10.1016/0022-1236(91)90099-Q.

[5]

L. CaffarelliB. Gidas and J. Spruck, Asymptotic symmetry and local behavior semilinear elliptic equations with critical Sobolev groth, Comm. Pure Appl. Math., 42 (1989), 271-297.  doi: 10.1002/cpa.3160420304.

[6]

W. ChenW. JinC. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalitities and systems of integral equations, Discrete Contin. Dyn. Syst. Suppl., 14 (2005), 164-172. 

[7]

C. Chen and C. Lin, Uniqueness of the ground state solution of $\Delta u+f(u)$ in $\mathbb{R}^n$, $n\geq3$, Commun. Partial Diff. Equ., 16 (1991), 1549-1572.  doi: 10.1080/03605309108820811.

[8]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.  doi: 10.1215/S0012-7094-91-06325-8.

[9]

W. Chen and C. Li, On Nirenberg and related problemsa necessary and sufficient condition, Commun. Pure Appl. Math., 48 (1995), 657-667.  doi: 10.1002/cpa.3160480606.

[10]

L. ChenZ. Liu and G. Lu, Symmetry and Regularity of solutions to the Weighted Hardy-Sobolev Type System, Adv. Nonlinear Stud., 16 (2016), 1-13.  doi: 10.1515/ans-2015-5005.

[11]

W. ChenC. Li and B. Ou, Classification of solutions for a systen of integral equations, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[12]

C. Coffman, On the positive solutions of boundary value problems for a class of nonlinear differential equatins, J. Diff. Eq., 3 (1967), 92-111.  doi: 10.1016/0022-0396(67)90009-5.

[13]

W. Dai, J. Huang, Y. Qin, B. Wang and Y. Fang, Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete Contin. Dyn. Syst., 39 (2019), 1389C–1403. doi: 10.3934/dcds.2018117.

[14]

Y. DingF. Gao and M. Yang, Semiclassical states for Choquard type equations with critical growth: Critical frequency case, Nonlinearity, 33 (2020), 6695-6728.  doi: 10.1088/1361-6544/aba88d.

[15]

L. Du, F. Gao and M. Yang, Existence and qualitative analysis for nonlinear weighted Choqaurd equations, preprint, arXiv: 1810.11759.

[16]

L. Du and M. Yang, Uniqueness and nondegeneracy of solutions for a critical nonlocal equation, Discrete Contin. Dyn. Syst., 39 (2019), 5847-5866.  doi: 10.3934/dcds.2019219.

[17]

F. GaoE. SilvaM. Yang and J. Zhou, Existence of solutions for critical Choquard equations via the concentration compactness method, Proc. Roy. Soc. Edinb. A, 150 (2020), 921-954.  doi: 10.1017/prm.2018.131.

[18]

F. Gao, M. Yang and J. Zhou, Existence of multiple semiclassical solutions for a critical Choquard equation with indefinite potential, Nonlinear Anal. TMA, 195 (2020), 111817. doi: 10.1016/j.na.2020.111817.

[19]

J. Giacomoni, Y. Wei and M. Yang, Nondegeneracy of solutions for a critical Hartree equation, Nonlinear Anal. TMA, 2020, 111969. doi: 10.1016/j.na.2020.111969.

[20]

B. GidasW. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^N$, Adv. Math. Sppl. Stud., A (1981), 369-402. 

[21]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.

[22]

L. Guo, T. Hu, S. Peng and W. Shuai, Existence and uniqueness of solutions for Choquard equation involving Hard-Littlewood-Sobolev critical exponent, Calc. Var. partial Diff. Equ., 58 (2019), 128, 34 pp. doi: 10.1007/s00526-019-1585-1.

[23]

C. Jin and C. Li, Qualitative analysis of some systems of integral equations, Calc. Var. Partial Differential Equations, 26 (2006), 447-457.  doi: 10.1007/s00526-006-0013-5.

[24]

H. Kaper and M. Kwong., Uniqueness of non-negative solutions of a class of semi-linear elliptic equations, in Nonlinear Diffusion Equations and Their Equilibrium States, Springer-Verlag, New York, 1988, 1-17. doi: 10.1007/978-1-4612-0873-0.

[25]

M. Kwong, Uniqueness of positive solutions of $\Delta u+u^p = 0\mathbb{R}^n$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.

[26]

Y. Lei, Qualitative analysis for the Hartree-type equations, SIAM. J. Math. Anal., 45 (2013), 388-406.  doi: 10.1137/120879282.

[27]

C. Li, Local asymptotic symmetry of singular solutions of to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/s002220050023.

[28]

Y. Li, Remark on some confomally invariant integral equations: The method of moving spheres, J. Eur.Math.Soc., 2 (2004), 153-180. 

[29]

Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8.

[30]

E. H. Lieb, Existence and uniquenss of the minimizing solution of choquard's nonlinear equation, Stud. Appl. Math., 57 (1976/1977), 93-105.  doi: 10.1002/sapm197757293.

[31]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.

[32]

E. H. Lieb and M. Loss, Graduate Studies in Mathematics, 2$^{nd}$ edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1007/978-1-4612-0873-0.

[33]

P. L Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.

[34]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.

[35]

C. MaW. Chen and C. Li, Rugularity of solutions for an integral system of Wolff type, Advances in Mathematics, 226 (2011), 2676-2699.  doi: 10.1016/j.aim.2010.07.020.

[36]

K. Mcleod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u) = 0$ in $\mathbb{R}^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145.  doi: 10.1007/BF00275874.

[37]

I. M. MorozR. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.

[38]

V. Moroz and J. Van Schaftingen, Ground states of nonlinear choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.

[39]

V. Moroz and J. Van Schaftingen, Groundstates of nonlinear choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12. doi: 10.1142/S0219199715500054.

[40]

W. Ni and R. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of $\Delta u+f(u, r) = 0$, Comm. Pure and Appl. Math., 38 (1985), 69-108.  doi: 10.1002/cpa.3160380105.

[41]

P. Padilla, On Some Nonlinear Elliptic Equations, Ph.D Thesis, Courant Institute, 1994.

[42]

S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, (1954). doi: 10.1007/978-1-4612-0873-0.

[43]

L. Peletier and J. Serrin, Uniqueness of solutions of semilinear equations in $\mathbb{R}^n$, J. Diff. Eq., 61 (1986), 380-397.  doi: 10.1016/0022-0396(86)90112-9.

[44]

J. Seok, Limit profiles and uniqueness of ground states to the nonliner Choquard equations, Advances in Nonlinear Analysis, 20 (2019), 207-228.  doi: 10.1515/anona-2017-0182.

[45]

J. Serrin and M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J., 49 (2000), 897-923.  doi: 10.1512/iumj.2000.49.1893.

[46]

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura. Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.

[47]

P. Tod and I. M. Moroz, An analytical approach to the Schrödinger-Newton equations, Nonlinearity, 12 (1999), 201-216.  doi: 10.1088/0951-7715/12/2/002.

[48]

J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905. doi: 10.1063/1.3060169.

[49]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.  doi: 10.1007/s002080050258.

[50]

M. Yang and X. Zhou, On a Coupled Schödinger System with Stein-Weiss Type Convolution Part, J. Geom. Anal., (2021). doi: 10.1007/s12220-021-00645-w.

[51]

L. Zhang and C. Lin, Uniqueness of ground state solutions, Acta Math. Sci., 8 (1988), 449-468.  doi: 10.1016/S0252-9602(18)30321-7.

[52]

Y. ZhenF. GaoZ. Shen and M. Yang, On a class of coupled critical Hartree system with deepening potential, Math. Meth. Appl. Sci., 44 (2021), 772-798.  doi: 10.1002/mma.6785.

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