# American Institute of Mathematical Sciences

April  2021, 20(4): 1319-1345. doi: 10.3934/cpaa.2021022

## Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents

 a. School of Mathematical Sciences, Qufu Normal University, Shandong, 273165, China b. Department of Mathematics, School of Sciences, North University of China, Shanxi, 030051, China

* Corresponding author

Received  May 2020 Revised  December 2020 Published  April 2021 Early access  April 2021

Fund Project: This research was partially supported by the NSFC(11571197, ZR2020MA005)

We consider the following Choquard equation
 $\label{modelv11} \begin{cases} -(a\!+\!\varepsilon \int_{\Omega}|\nabla u|^2)\Delta u\! = \!\left( \int_{\Omega}\frac{|u(y)|^{2^{*}_{\mu}}}{|x-y|^\mu}dy\right)|u|^{2^{*}_{\mu}-2}u \!+\! \lambda f(x)|u|^{q-2}u \quad in \quad \Omega,\\ u\! = \!0 \qquad \qquad \qquad \qquad \qquad on \quad \partial\Omega, \end{cases}$
where
 $\lambda$
is a real parameter,
 $2^{*}_{\mu} = \frac{2N-\mu}{N-2}(0<\mu is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under some suitable assumptions on $ \lambda, \; \mu $, via the constrained minimizer method and concentration compactness principle, we prove that this system has multiple of solutions, and one of which is a positive ground state solution. Moreover, by using an abstract result due to K.-C Chang, we admit infinitely many pairs of distinct solutions. In addition, we prove the nonexistence result by Pohožaev identity when $ \lambda<0 $. The main results extend and complement the earlier works in the literature. Citation: Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1319-1345. doi: 10.3934/cpaa.2021022 ##### References:  [1] C. Alves, G. Figueiredo and M. Yang, Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity, Adv. Nonlinear Anal., 4 (2016), 331-345. doi: 10.1515/anona-2015-0123. [2] C. Alves, D. Cassani, C. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in$\mathbb{R}^2$, J. Differ. Equ., 261 (2016), 1933-1972. doi: 10.1016/j.jde.2016.04.021. [3] A. Ambrosetti, H. Brézis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078. [4] R. Arora, J. Giacomoni, T. Mukherjee and K. Sreenadh, Polyharmonic Kirchhoff problems involving exponential non-linearity of Choquard type with singular weights, Nonlinear Anal., 196 (2020), 1-24. doi: 10.1016/j.na.2020.111779. [5] L. Battaglia and J. Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations in the plane, Adv. Nonlinear Stud., 17 (2017), 581-594. doi: 10.1515/ans-2016-0038. [6] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. [7] D. Cassani and J. Zhang, Choquard-type equations with Hardy-Littlewood-Sobolev upper-critical growth, Adv. Nonlinear Anal., 8 (2019), 1184-1212. doi: 10.1515/anona-2018-0019. [8] M. Clapp and D. Salazar, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407 (2013), 1-15. doi: 10.1016/j.jmaa.2013.04.081. [9] K. Chang, Methods in Nonlinear Analysis, Springer-Verlag, Berlin, 2005. [10] S. Chen, B. Zhang and X. Tang, Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity, Adv. Nonlinear Anal., 9 (2018), 148-167. doi: 10.1515/anona-2018-0147. [11] F. Gao, E. Silva, M. Yang and J. Zhou, Existence of solutions for critical Choquard equations via the concentration compactness method, P. Roy. Soc. Edinb. A., 150 (2020), 921-954. doi: 10.1017/prm.2018.131. [12] F. Gao and M. Yang, The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation, Sci. China Math., 61 (2018), 1219-1242. doi: 10.1007/s11425-016-9067-5. [13] D. Goel and K Sreenadh, Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity, Nonlinear Anal., 186 (2019), 162-186. doi: 10.1016/j.na.2019.01.035. [14] C. Lei, G. Liu and L. Gao, Multiple positive solutions for Kirchhoff type problem with a critical nonlinearity, Nonlinear Anal., 31 (2016), 343-355. doi: 10.1016/j.nonrwa.2016.01.018. [15] G. Li and C. Tang, Existence of a ground state solution for Choquard equation with the upper critical exponent, Comput. Math. Appl., 76 (2018), 2635-2647. doi: 10.1016/j.camwa.2018.08.052. [16] F. Li, C. Gao and X. Zhu, Existence and concentration of sign-changing solutions to Kirchhoff type system with Hartree-type nonlinearity, J. Math. Anal. Appl., 448 (2017), 60-80. doi: 10.1016/j.jmaa.2016.10.069. [17] J. Liao, H. Li and P. Zhang, Existence and multiplicity of solutions for a nonnlcal problem with critical Sobolev exponent, Comput. Math. Appl., 75 (2018), 787-797. doi: 10.1016/j.camwa.2017.10.012. [18] E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1976/77), 93-105. doi: 10.1002/sapm197757293. [19] E. Lieb and M. Loss, Analysis, Graduate Studies Mathematics, AMS, Providence, Rhode Island, 2001. [20] P. Lions, The concentration-compactness principle in the calculus of variations, The limit case, Rev. Mat. Iberoam., 1 (1985), 145-201. doi: 10.4171/RMI/6. [21] V. Moroz and J. Schaftingen, Groundstate of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 152-184. doi: 10.1016/j.jfa.2013.04.007. [22] V. Moroz and J. Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579. doi: 10.1090/S0002-9947-2014-06289-2. [23] V. Moroz and J. Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1-12. doi: 10.1142/S0219199715500054. [24] T. Mukherjee and K. Sreenadh, Fractional Choquard equation with critical nonlinearities, Nolinear Differ. Equ. Appl., 24 (2017), 1-34. doi: 10.1007/s00030-017-0487-1. [25] S. Pekar, Untersuchungber die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. [26] P. Pucci, M. Xiang and B. Zhang, Existence results for Schrödinger-Choquard-Kirchhoff equations involving the fractional p-Laplacian, Adv. Calc. Var., 12 (2019), 253-275. doi: 10.1515/acv-2016-0049. [27] M. Willem, Minimax Theorems, Birthäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [28] M. Xiang, D. Rădulescu and B. Zhang, A critical fractional Choquard-Kirchhoff problem with magnetic field, Commun. Contemp. Math., 21 (2019), 1-36. doi: 10.1142/s0219199718500049. show all references ##### References:  [1] C. Alves, G. Figueiredo and M. Yang, Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity, Adv. Nonlinear Anal., 4 (2016), 331-345. doi: 10.1515/anona-2015-0123. [2] C. Alves, D. Cassani, C. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in$\mathbb{R}^2$, J. Differ. Equ., 261 (2016), 1933-1972. doi: 10.1016/j.jde.2016.04.021. [3] A. Ambrosetti, H. Brézis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078. [4] R. Arora, J. Giacomoni, T. Mukherjee and K. Sreenadh, Polyharmonic Kirchhoff problems involving exponential non-linearity of Choquard type with singular weights, Nonlinear Anal., 196 (2020), 1-24. doi: 10.1016/j.na.2020.111779. [5] L. Battaglia and J. Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations in the plane, Adv. Nonlinear Stud., 17 (2017), 581-594. doi: 10.1515/ans-2016-0038. [6] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. [7] D. Cassani and J. Zhang, Choquard-type equations with Hardy-Littlewood-Sobolev upper-critical growth, Adv. Nonlinear Anal., 8 (2019), 1184-1212. doi: 10.1515/anona-2018-0019. [8] M. Clapp and D. Salazar, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407 (2013), 1-15. doi: 10.1016/j.jmaa.2013.04.081. [9] K. Chang, Methods in Nonlinear Analysis, Springer-Verlag, Berlin, 2005. [10] S. Chen, B. Zhang and X. Tang, Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity, Adv. Nonlinear Anal., 9 (2018), 148-167. doi: 10.1515/anona-2018-0147. [11] F. Gao, E. Silva, M. Yang and J. Zhou, Existence of solutions for critical Choquard equations via the concentration compactness method, P. Roy. Soc. Edinb. A., 150 (2020), 921-954. doi: 10.1017/prm.2018.131. [12] F. Gao and M. Yang, The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation, Sci. China Math., 61 (2018), 1219-1242. doi: 10.1007/s11425-016-9067-5. [13] D. Goel and K Sreenadh, Kirchhoff equations with Hardy-Littlewood-Sobolev critical nonlinearity, Nonlinear Anal., 186 (2019), 162-186. doi: 10.1016/j.na.2019.01.035. [14] C. Lei, G. Liu and L. Gao, Multiple positive solutions for Kirchhoff type problem with a critical nonlinearity, Nonlinear Anal., 31 (2016), 343-355. doi: 10.1016/j.nonrwa.2016.01.018. [15] G. Li and C. Tang, Existence of a ground state solution for Choquard equation with the upper critical exponent, Comput. Math. Appl., 76 (2018), 2635-2647. doi: 10.1016/j.camwa.2018.08.052. [16] F. Li, C. Gao and X. Zhu, Existence and concentration of sign-changing solutions to Kirchhoff type system with Hartree-type nonlinearity, J. Math. Anal. Appl., 448 (2017), 60-80. doi: 10.1016/j.jmaa.2016.10.069. [17] J. Liao, H. Li and P. Zhang, Existence and multiplicity of solutions for a nonnlcal problem with critical Sobolev exponent, Comput. Math. Appl., 75 (2018), 787-797. doi: 10.1016/j.camwa.2017.10.012. [18] E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1976/77), 93-105. doi: 10.1002/sapm197757293. [19] E. Lieb and M. Loss, Analysis, Graduate Studies Mathematics, AMS, Providence, Rhode Island, 2001. [20] P. Lions, The concentration-compactness principle in the calculus of variations, The limit case, Rev. Mat. Iberoam., 1 (1985), 145-201. doi: 10.4171/RMI/6. [21] V. Moroz and J. Schaftingen, Groundstate of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 152-184. doi: 10.1016/j.jfa.2013.04.007. [22] V. Moroz and J. Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579. doi: 10.1090/S0002-9947-2014-06289-2. [23] V. Moroz and J. Schaftingen, Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1-12. doi: 10.1142/S0219199715500054. [24] T. Mukherjee and K. Sreenadh, Fractional Choquard equation with critical nonlinearities, Nolinear Differ. Equ. Appl., 24 (2017), 1-34. doi: 10.1007/s00030-017-0487-1. [25] S. Pekar, Untersuchungber die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. [26] P. Pucci, M. Xiang and B. 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