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

Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents

Xiaorong Luo a, , Anmin Mao a,, and  Yanbin Sang a,b,

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<N) $
 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.
Keywords: Hardy-Littlewood-Sobolev inequality, Choquard equations, critical exponent, variational method.
Mathematics Subject Classification: Primary: 35J05, 35J20, 35J60.
Citation: Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[9]

K. Chang, Methods in Nonlinear Analysis, Springer-Verlag, Berlin, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

E. Lieb and M. Loss, Analysis, Graduate Studies Mathematics, AMS, Providence, Rhode Island, 2001. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

S. Pekar, Untersuchungber die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. Google Scholar

[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.  Google Scholar

[27]

M. Willem, Minimax Theorems, Birthäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

K. Chang, Methods in Nonlinear Analysis, Springer-Verlag, Berlin, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

E. Lieb and M. Loss, Analysis, Graduate Studies Mathematics, AMS, Providence, Rhode Island, 2001. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

S. Pekar, Untersuchungber die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. Google Scholar

[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.  Google Scholar

[27]

M. Willem, Minimax Theorems, Birthäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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.  Google Scholar

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