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October  2019, 39(10): 5847-5866. doi: 10.3934/dcds.2019219

## Uniqueness and nondegeneracy of solutions for a critical nonlocal equation

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

* Corresponding author: Minbo Yang, he was partially supported by NSFC (11571317) and ZJNSF(LD19A010001)

Received  October 2018 Revised  February 2019 Published  July 2019

The aim of this paper is to classify the positive solutions of the nonlocal critical equation:
 $- \Delta u = \left( {{I_\mu }*{u^{2_\mu ^*}}} \right){u^{2_\mu ^* - 1}},x \in {{\mathbb{R}}^N}$
where
 $0<\mu , if $ N = 3\ \hbox{or} \ 4 $and $ 0<\mu\leq4 $if $ N\geq5 $, $ I_{\mu} $is the Riesz potential defined by ${I_\mu }(x) = \frac{{\Gamma \left( {\frac{\mu }{2}} \right)}}{{\Gamma \left( {\frac{{N - \mu }}{2}} \right){\pi ^{\frac{N}{2}}}{2^{N - \mu }}|x{|^\mu }}}$with $ \Gamma(s) = \int^{+\infty}_{0}x^{s-1}e^{-x}dx $, $ s>0 $and $ 2^{\ast}_{\mu} = \frac{2N-\mu}{N-2} $is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. We apply the moving plane method in integral forms to prove the symmetry and uniqueness of the positive solutions. Moreover, we also prove the nondegeneracy of the unique solutions for the equation when $ \mu $close to $ N $. Citation: Lele Du, Minbo Yang. Uniqueness and nondegeneracy of solutions for a critical nonlocal equation. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5847-5866. doi: 10.3934/dcds.2019219 ##### References:  [1] C. O. Alves, Fa shun Gao, M. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differential Equations, 263 (2017), 3943-3988. doi: 10.1016/j.jde.2017.05.009. [2] T. Aubin, Best constants in the Sobolev imbedding theorem: The Yamabe problem, Ann. of Math. Stud., 102 (1982), 173-184. [3] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. [4] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. [5] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [6] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354. doi: 10.3934/dcds.2005.12.347. [7] 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. [8] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, , AIMS Ser. Differ. Equ. Dyn. Syst., vol.4, 2010. [9] W. Chen and C. Li, Regularity of solutions for a system of integral equations, Commun. Pure Appl. Anal., 4 (2005), 1-8. doi: 10.3934/cpaa.2005.4.1. [10] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038. [11] W. Chen, C. Li and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157. doi: 10.1016/j.jfa.2017.02.022. [12] F. Gao, E. da Silva, M. Yang and J. Zhou, Existence of solutions for critical Choquard equations via the concentration compactness method, Proc. Roy. Soc. Edinb. A, 2019. doi: 10.1017/prm.2018.131. [13] F. Gao and M. Yang, A strongly indefinite Choquard equation with critical exponent due to the Hardy-Littlewood-Sobolev inequality, Commun.Contemp. Math., 20 (2018), 1750037, 22 pp. doi: 10.1142/S0219199717500377. [14] F. Gao and M. Yang, The Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci. China Math., 61 (2018), 1219-1242. doi: 10.1007/s11425-016-9067-5. [15] F. Gao and M. Yang, On nonlocal Choquard equations with Hardy–Littlewood–Sobolev critical exponents, Journal of Mathematical Analysis and Applications, 448 (2017), 1006-1041. doi: 10.1016/j.jmaa.2016.11.015. [16] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in$\mathbb{R}^{N}$, Mathematical Analysis and Applications, 7 (1981), 369-402. [17] 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. [18] N. S. Landkof, Foundations of Modern Potential Theory, translated by A. P. Doohovskoy, Grundlehren der mathematischen Wissenschaften, Springer, New York-Heidelberg, 1972. [19] Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273 (2013), 883-905. doi: 10.1007/s00209-012-1036-6. [20] Y. Lei, Qualitative analysis for the Hartree-type equations, SIAM J. Math. Anal., 45 (2013), 388-406. doi: 10.1137/120879282. [21] Y. Lei, Liouville theorems and classification results for a nonlocal schrodinger equation, Discrete Contin. Dyn. Syst. A, 38 (2018), 5351-5377. doi: 10.3934/dcds.2018236. [22] Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45 (2012), 43-61. doi: 10.1007/s00526-011-0450-7. [23] Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. [24] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/BF01232373. [25] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. [26] E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105. doi: 10.1002/sapm197757293. [27] E. Lieb and M. Loss, Analysis, , Gradute Studies in Mathematics, 1997. doi: 10.1090/gsm/014. [28] P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4. [29] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 1796-1806. doi: 10.1016/j.na.2009.01.014. [30] G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. Partial Differential Equations, 42 (2011), 563-577. doi: 10.1007/s00526-011-0398-7. [31] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Ration. Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3. [32] V. Moroz and J. Van Schaftingen, Groundstates 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. [33] J. Seok, Limit profiles and uniqueness of ground states to the nonlinear Choquard equations, Adv. Nonlinear Anal., 2018. doi: 10.1515/anona-2017-0182. [34] Z. Shen, F. Gao and M. Yang, On critical Choquard equation with potential well, Discrete Contin. Dyn. Syst. A., 38 (2018), 3669-3695. doi: 10.3934/dcds.2018151. [35] Z. Shen, F. Gao and M. Yang, Multiple solutions for nonhomogeneous Choquard equation involving Hardy-Littlewood-Sobolev critical exponent, Z. Angew. Math. Phys., 68 (2017), Art. 61, 25 pp. doi: 10.1007/s00033-017-0806-8. [36] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura. Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013. show all references ##### References:  [1] C. O. Alves, Fa shun Gao, M. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differential Equations, 263 (2017), 3943-3988. doi: 10.1016/j.jde.2017.05.009. [2] T. Aubin, Best constants in the Sobolev imbedding theorem: The Yamabe problem, Ann. of Math. Stud., 102 (1982), 173-184. [3] L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297. doi: 10.1002/cpa.3160420304. [4] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116. [5] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445. [6] W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354. doi: 10.3934/dcds.2005.12.347. [7] 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. [8] W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, , AIMS Ser. Differ. Equ. Dyn. Syst., vol.4, 2010. [9] W. Chen and C. Li, Regularity of solutions for a system of integral equations, Commun. Pure Appl. Anal., 4 (2005), 1-8. doi: 10.3934/cpaa.2005.4.1. [10] W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038. [11] W. Chen, C. Li and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157. doi: 10.1016/j.jfa.2017.02.022. [12] F. Gao, E. da Silva, M. Yang and J. Zhou, Existence of solutions for critical Choquard equations via the concentration compactness method, Proc. Roy. Soc. Edinb. A, 2019. doi: 10.1017/prm.2018.131. [13] F. Gao and M. Yang, A strongly indefinite Choquard equation with critical exponent due to the Hardy-Littlewood-Sobolev inequality, Commun.Contemp. Math., 20 (2018), 1750037, 22 pp. doi: 10.1142/S0219199717500377. [14] F. Gao and M. Yang, The Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci. China Math., 61 (2018), 1219-1242. doi: 10.1007/s11425-016-9067-5. [15] F. Gao and M. Yang, On nonlocal Choquard equations with Hardy–Littlewood–Sobolev critical exponents, Journal of Mathematical Analysis and Applications, 448 (2017), 1006-1041. doi: 10.1016/j.jmaa.2016.11.015. [16] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in$\mathbb{R}^{N}$, Mathematical Analysis and Applications, 7 (1981), 369-402. [17] 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. [18] N. S. Landkof, Foundations of Modern Potential Theory, translated by A. P. Doohovskoy, Grundlehren der mathematischen Wissenschaften, Springer, New York-Heidelberg, 1972. [19] Y. Lei, On the regularity of positive solutions of a class of Choquard type equations, Math. Z., 273 (2013), 883-905. doi: 10.1007/s00209-012-1036-6. [20] Y. Lei, Qualitative analysis for the Hartree-type equations, SIAM J. Math. Anal., 45 (2013), 388-406. doi: 10.1137/120879282. [21] Y. Lei, Liouville theorems and classification results for a nonlocal schrodinger equation, Discrete Contin. Dyn. Syst. A, 38 (2018), 5351-5377. doi: 10.3934/dcds.2018236. [22] Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system, Calc. Var. Partial Differential Equations, 45 (2012), 43-61. doi: 10.1007/s00526-011-0450-7. [23] Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180. [24] C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231. doi: 10.1007/BF01232373. [25] E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032. [26] E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105. doi: 10.1002/sapm197757293. [27] E. Lieb and M. Loss, Analysis, , Gradute Studies in Mathematics, 1997. doi: 10.1090/gsm/014. [28] P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4. [29] S. Liu, Regularity, symmetry, and uniqueness of some integral type quasilinear equations, Nonlinear Anal., 71 (2009), 1796-1806. doi: 10.1016/j.na.2009.01.014. [30] G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. Partial Differential Equations, 42 (2011), 563-577. doi: 10.1007/s00526-011-0398-7. [31] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Ration. Mech. Anal., 195 (2010), 455-467. doi: 10.1007/s00205-008-0208-3. [32] V. Moroz and J. Van Schaftingen, Groundstates 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. [33] J. Seok, Limit profiles and uniqueness of ground states to the nonlinear Choquard equations, Adv. Nonlinear Anal., 2018. doi: 10.1515/anona-2017-0182. [34] Z. Shen, F. Gao and M. Yang, On critical Choquard equation with potential well, Discrete Contin. Dyn. Syst. A., 38 (2018), 3669-3695. doi: 10.3934/dcds.2018151. [35] Z. Shen, F. Gao and M. Yang, Multiple solutions for nonhomogeneous Choquard equation involving Hardy-Littlewood-Sobolev critical exponent, Z. Angew. Math. Phys., 68 (2017), Art. 61, 25 pp. doi: 10.1007/s00033-017-0806-8. [36] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura. Appl., 110 (1976), 353-372. doi: 10.1007/BF02418013.  [1] Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265 [2] Zaizheng Li, Zhitao Zhang. 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