# American Institute of Mathematical Sciences

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  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
 $$$-\Delta u = (I_{\mu_1}\ast|u|^{2_{\mu_1}^\ast})|u|^{2_{\mu_1}^\ast-2}u +(I_{\mu_2}\ast|u|^{2_{\mu_2}^\ast})|u|^{2_{\mu_2}^\ast-2}u,\; x\in\mathbb{R}^N, \;\;\;\;\;\;\;(1)$$$
where
 $0<\mu_1,\mu_2 if $ N = 3 $or $ 4 $, and $ N-4\leq\mu_1,\mu_2
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 & Continuous Dynamical Systems, doi: 10.3934/dcds.2021074
##### References:
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Stud., A (1981), 369-402.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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 doi: 10.1007/978-1-4612-0873-0.  Google Scholar [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.  Google Scholar [26] Y. Lei, Qualitative analysis for the Hartree-type equations, SIAM. J. Math. Anal., 45 (2013), 388-406.  doi: 10.1137/120879282.  Google Scholar [27] C. Li, Local asymptotic symmetry of singular solutions of to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/s002220050023.  Google Scholar [28] Y. Li, Remark on some confomally invariant integral equations: The method of moving spheres, J. Eur.Math.Soc., 2 (2004), 153-180.   Google Scholar [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.  Google Scholar [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.  Google Scholar [31] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar [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.  Google Scholar [33] 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 [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.  Google Scholar [35] C. Ma, W. 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.  Google Scholar [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.  Google Scholar [37] I. M. Moroz, R. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [41] P. Padilla, On Some Nonlinear Elliptic Equations, Ph.D Thesis, Courant Institute, 1994.  Google Scholar [42] S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, (1954). doi: 10.1007/978-1-4612-0873-0.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [46] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura. Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [52] Y. Zhen, F. Gao, Z. 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.  Google Scholar

show all references

##### References:
 [1] A. Alexandrov, Uniqueness theorems for surfaces in large V, Am.of Math. Soc.Transl, 21 (1962), 412-416.   Google Scholar [2] T. Aubin, Best constans in the Sobolev imbedding theorem: the Yamabe problem, Ann.of Math. Stud., 115 (1989), 173-184.   Google Scholar [3] T. Bartch, T. 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.  Google Scholar [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.  Google Scholar [5] L. Caffarelli, B. 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.  Google Scholar [6] W. Chen, W. Jin, C. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalitities and systems of integral equations, Discrete Contin. Dyn. Syst. Suppl., 14 (2005), 164-172.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] L. Chen, Z. 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.  Google Scholar [11] W. Chen, C. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] Y. Ding, F. 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.  Google Scholar [15] L. Du, F. Gao and M. Yang, Existence and qualitative analysis for nonlinear weighted Choqaurd equations, preprint, arXiv: 1810.11759. Google Scholar [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.  Google Scholar [17] F. Gao, E. Silva, M. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] B. Gidas, W. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^N$, Adv. Math. Sppl. Stud., A (1981), 369-402.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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 doi: 10.1007/978-1-4612-0873-0.  Google Scholar [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.  Google Scholar [26] Y. Lei, Qualitative analysis for the Hartree-type equations, SIAM. J. Math. Anal., 45 (2013), 388-406.  doi: 10.1137/120879282.  Google Scholar [27] C. Li, Local asymptotic symmetry of singular solutions of to nonlinear elliptic equations, Invent. Math., 123 (1996), 221-231.  doi: 10.1007/s002220050023.  Google Scholar [28] Y. Li, Remark on some confomally invariant integral equations: The method of moving spheres, J. Eur.Math.Soc., 2 (2004), 153-180.   Google Scholar [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.  Google Scholar [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.  Google Scholar [31] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar [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.  Google Scholar [33] 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 [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.  Google Scholar [35] C. Ma, W. 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.  Google Scholar [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.  Google Scholar [37] I. M. Moroz, R. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [41] P. Padilla, On Some Nonlinear Elliptic Equations, Ph.D Thesis, Courant Institute, 1994.  Google Scholar [42] S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, (1954). doi: 10.1007/978-1-4612-0873-0.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [46] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura. 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