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<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.
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. AlvesG. 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. AlvesD. CassaniC. 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

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A. AmbrosettiH. 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. AroraJ. GiacomoniT. 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. GaoE. SilvaM. 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. LeiG. 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. LiC. 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. LiaoH. 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. PucciM. 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. XiangD. 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. AlvesG. 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. AlvesD. CassaniC. 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. AmbrosettiH. 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. AroraJ. GiacomoniT. 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. GaoE. SilvaM. 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. LeiG. 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. LiC. 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. LiaoH. 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. PucciM. 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. XiangD. 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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