# American Institute of Mathematical Sciences

• Previous Article
Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition
• CPAA Home
• This Issue
• Next Article
Partial regularity for parabolic systems with VMO-coefficients
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

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 $. The main results extend and complement the earlier works in the literature. Citation: Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021022 ##### References:  [1] C. Alves, G. Figueiredo and M. Yang, Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity, Adv. Nonlinear Anal., 4 (2016), 331-345. doi: 10.1515/anona-2015-0123. Google Scholar [2] C. Alves, D. Cassani, C. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in$\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 [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 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  [1] Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 [2] Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 [3] Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021038 [4] Claudianor O. Alves, Giovany M. Figueiredo, Riccardo Molle. Multiple positive bound state solutions for a critical Choquard equation. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021061 [5] Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827 [6] Grace Nnennaya Ogwo, Chinedu Izuchukwu, Oluwatosin Temitope Mewomo. A modified extragradient algorithm for a certain class of split pseudo-monotone variational inequality problem. Numerical Algebra, Control & Optimization, 2021 doi: 10.3934/naco.2021011 [7] Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209 [8] Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 [9] Kazuhiro Kurata, Yuki Osada. Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021100 [10] Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021016 [11] Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on$ C^{1} $domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002 [12] Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018 [13] Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 [14] Nicolas Dirr, Hubertus Grillmeier, Günther Grün. On stochastic porous-medium equations with critical-growth conservative multiplicative noise. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2829-2871. doi: 10.3934/dcds.2020388 [15] Bruno Premoselli. Einstein-Lichnerowicz type singular perturbations of critical nonlinear elliptic equations in dimension 3. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021069 [16] Peng Chen, Xiaochun Liu. Positive solutions for Choquard equation in exterior domains. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021065 [17] Jiangang Qi, Bing Xie. Extremum estimates of the$ L^1 \$-norm of weights for eigenvalue problems of vibrating string equations based on critical equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3505-3516. doi: 10.3934/dcdsb.2020243 [18] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [19] Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222 [20] Derrick Jones, Xu Zhang. A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces. Electronic Research Archive, , () : -. doi: 10.3934/era.2021032

2019 Impact Factor: 1.105

Article outline