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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<N $
, 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 & Continuous Dynamical Systems - A, 2019, 39 (10) : 5847-5866. doi: 10.3934/dcds.2019219
References:
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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.  Google Scholar

[22]

Y. LeiC. 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.  Google Scholar

[23]

Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.   Google Scholar

[24]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/BF01232373.  Google Scholar

[25]

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

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

Z. ShenF. 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.  Google Scholar

[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.  Google Scholar

[36]

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

show all references

References:
[1]

C. O. AlvesFa shun GaoM. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differential Equations, 263 (2017), 3943-3988.  doi: 10.1016/j.jde.2017.05.009.  Google Scholar

[2]

T. Aubin, Best constants in the Sobolev imbedding theorem: The Yamabe problem, Ann. of Math. Stud., 102 (1982), 173-184.   Google Scholar

[3]

L. CaffarelliB. 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.  Google Scholar

[4]

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

[5]

W. ChenC. 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.  Google Scholar

[6]

W. ChenC. 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.  Google Scholar

[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.  Google Scholar

[8]

W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, , AIMS Ser. Differ. Equ. Dyn. Syst., vol.4, 2010.  Google Scholar

[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.  Google Scholar

[10]

W. ChenC. 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.  Google Scholar

[11]

W. ChenC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

B. GidasW. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, Mathematical Analysis and Applications, 7 (1981), 369-402.   Google Scholar

[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.  Google Scholar

[18]

N. S. Landkof, Foundations of Modern Potential Theory, translated by A. P. Doohovskoy, Grundlehren der mathematischen Wissenschaften, Springer, New York-Heidelberg, 1972.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

Y. LeiC. 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.  Google Scholar

[23]

Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.   Google Scholar

[24]

C. Li, Local asymptotic symmetry of singular solutions to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/BF01232373.  Google Scholar

[25]

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

[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.  Google Scholar

[27]

E. Lieb and M. Loss, Analysis, , Gradute Studies in Mathematics, 1997. doi: 10.1090/gsm/014.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

Z. ShenF. 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.  Google Scholar

[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.  Google Scholar

[36]

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

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