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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 |
$ \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} $ |
$ \lambda $ |
$ 2^{*}_{\mu} = \frac{2N-\mu}{N-2}(0<\mu<N) $ |
$ \lambda, \; \mu $ |
$ \lambda<0 $ |
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
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