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The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations
School of Mathematics, Southeast University, Nanjing 210096, China |
$ \begin{equation*} \left\{ \begin{array}{lr} -\Delta u = |u|^4u+\phi |u|^3 u +\lambda u,\quad& in\,\,\Omega,\\ -\Delta \phi = |u|^5,\quad& in\,\,\Omega,\\ u = \phi = 0,\quad& on\,\,\partial\Omega, \end{array} \right. \end{equation*} $ |
$ \Omega $ |
$ \mathbb{R}^3 $ |
$ \lambda $ |
References:
[1] |
C. O. Alves, A. B. Nóbrega and M. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differ. Equ., 55 (2016), Art. 48, 28 pp.
doi: 10.1007/s00526-016-0984-9. |
[2] |
C. O. Alves and M. Yang,
Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differ. Equations., 257 (2014), 4133-4164.
doi: 10.1016/j.jde.2014.08.004. |
[3] |
C. O. Alves and M. Yang,
Multiplicity and concentration of solutions for a quasilinear Choquard equation, J. Math. Phys., 55 (2014), 423-443.
doi: 10.1063/1.4884301. |
[4] |
Y. Ao, Existence of solutions for a class of nonlinear Choquard equations with critical growth, Appl. Anal., (2019), 1–17.
doi: 10.1080/00036811.2019.1608961. |
[5] |
A. Azzollini and P. d'Avenia,
On a system involving a critically growing nonlinearity, J. Math. Anal. Appl., 387 (2012), 433-438.
doi: 10.1016/j.jmaa.2011.09.012. |
[6] |
H. Brézis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pur. Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[7] |
L. Cai and F. Zhang, The Brezis-Nirenberg type double critical problem for the Choquard equation, SN Partial Differential Equations and Applications, 1 (2020).
doi: 10.1007/s42985-020-00032-0. |
[8] |
A. Capozzi, D. Fortunato and G. Palmieri,
An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 463-470.
doi: 10.1016/S0294-1449(16)30395-X. |
[9] |
G. Cerami, S. Solimini and M. Struwe,
Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal., 69 (1986), 289-306.
doi: 10.1016/0022-1236(86)90094-7. |
[10] |
S. Cingolani, M. Clapp and S. Secchi,
Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233-248.
doi: 10.1007/s00033-011-0166-8. |
[11] |
Xiaojing Feng, Ground state solution for a class of Schrödinger-Poisson-type systems with partial potential, Z. Angew. Math. Phys., 71 (2020), Paper No. 37, 16 pp.
doi: 10.1007/s00033-020-1254-4. |
[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] |
C. Y. Lei, G. S. Liu and H. M. Suo, Positive solutions for a Schrödinger-Poisson system with singularity and critical exponent, J. Math. Anal. Appl., 483 (2020), 123647, 21 pp.
doi: 10.1016/j.jmaa.2019.123647. |
[14] |
F. Li, Y. Li and J. Shi, Existence of positive solutions to Schrödinger-Poisson type systems with critical exponent, Commun. Contemp. Math., 16 (2014), 1450036, 28 pp.
doi: 10.1142/S0219199714500369. |
[15] |
F. Li, L. Long, Y. Huang and Z. Liang, Ground state for Choquard equation with doubly critical growth nonlinearity, J. Qual. Theory Differ. Equ., (2019), 1–15.
doi: 10.14232/ejqtde.2019.1.33. |
[16] |
X. Li and S. Ma,
Ground states for Choquard equations with doubly critical exponents, Rocky Mountain J. Math., 49 (2019), 153-170.
doi: 10.1216/RMJ-2019-49-1-153. |
[17] |
G. D. Li and C. L. 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. |
[18] |
E. H. Lieb,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.
doi: 10.1002/sapm197757293. |
[19] |
P.-L. Lions,
The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[20] |
L. Ma and L. Zhao,
Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.
doi: 10.1007/s00205-008-0208-3. |
[21] |
V. Moroz and J. Van Schaftingen,
Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct Anal., 265 (2013), 153-184.
doi: 10.1016/j.jfa.2013.04.007. |
[22] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, RI, 1986.
doi: 10.1090/cbms/065. |
[23] |
J. Seok,
Nonlinear Choquard equations: Doubly critical case, Appl. Math. Lett., 76 (2018), 148-156.
doi: 10.1016/j.aml.2017.08.016. |
[24] |
M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-662-02624-3. |
[25] |
J. Van Schaftingen and J. Xia,
Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent, J. Math. Anal. Appl., 464 (2018), 1184-1202.
doi: 10.1016/j.jmaa.2018.04.047. |
[26] |
M. Willem, Minimax Theorems, Birkhäuser Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[27] |
M. Willem, Functional Analysis, Springer New York, 2013.
doi: 10.1007/978-1-4614-7004-5. |
[28] |
M. B. Yang, J. C. de Albuquerque, E. D. Silva and M. L. Silva,
On the critical cases of linearly coupled Choquard systems, Appl. Math. Lett., 91 (2019), 1-8.
doi: 10.1016/j.aml.2018.11.005. |
[29] |
Qi Zhang,
Existence, uniqueness and multiplicity of positive solutions for schrödinger-Poisson system with singularity, J. Math. Anal. Appl., 437 (2016), 160-180.
doi: 10.1016/j.jmaa.2015.12.061. |
[30] |
F. Zhang and M. Du,
Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well, J. Differ. Equations, 269 (2020), 10085-10106.
doi: 10.1016/j.jde.2020.07.013. |
show all references
References:
[1] |
C. O. Alves, A. B. Nóbrega and M. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differ. Equ., 55 (2016), Art. 48, 28 pp.
doi: 10.1007/s00526-016-0984-9. |
[2] |
C. O. Alves and M. Yang,
Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differ. Equations., 257 (2014), 4133-4164.
doi: 10.1016/j.jde.2014.08.004. |
[3] |
C. O. Alves and M. Yang,
Multiplicity and concentration of solutions for a quasilinear Choquard equation, J. Math. Phys., 55 (2014), 423-443.
doi: 10.1063/1.4884301. |
[4] |
Y. Ao, Existence of solutions for a class of nonlinear Choquard equations with critical growth, Appl. Anal., (2019), 1–17.
doi: 10.1080/00036811.2019.1608961. |
[5] |
A. Azzollini and P. d'Avenia,
On a system involving a critically growing nonlinearity, J. Math. Anal. Appl., 387 (2012), 433-438.
doi: 10.1016/j.jmaa.2011.09.012. |
[6] |
H. Brézis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pur. Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[7] |
L. Cai and F. Zhang, The Brezis-Nirenberg type double critical problem for the Choquard equation, SN Partial Differential Equations and Applications, 1 (2020).
doi: 10.1007/s42985-020-00032-0. |
[8] |
A. Capozzi, D. Fortunato and G. Palmieri,
An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 463-470.
doi: 10.1016/S0294-1449(16)30395-X. |
[9] |
G. Cerami, S. Solimini and M. Struwe,
Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal., 69 (1986), 289-306.
doi: 10.1016/0022-1236(86)90094-7. |
[10] |
S. Cingolani, M. Clapp and S. Secchi,
Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233-248.
doi: 10.1007/s00033-011-0166-8. |
[11] |
Xiaojing Feng, Ground state solution for a class of Schrödinger-Poisson-type systems with partial potential, Z. Angew. Math. Phys., 71 (2020), Paper No. 37, 16 pp.
doi: 10.1007/s00033-020-1254-4. |
[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] |
C. Y. Lei, G. S. Liu and H. M. Suo, Positive solutions for a Schrödinger-Poisson system with singularity and critical exponent, J. Math. Anal. Appl., 483 (2020), 123647, 21 pp.
doi: 10.1016/j.jmaa.2019.123647. |
[14] |
F. Li, Y. Li and J. Shi, Existence of positive solutions to Schrödinger-Poisson type systems with critical exponent, Commun. Contemp. Math., 16 (2014), 1450036, 28 pp.
doi: 10.1142/S0219199714500369. |
[15] |
F. Li, L. Long, Y. Huang and Z. Liang, Ground state for Choquard equation with doubly critical growth nonlinearity, J. Qual. Theory Differ. Equ., (2019), 1–15.
doi: 10.14232/ejqtde.2019.1.33. |
[16] |
X. Li and S. Ma,
Ground states for Choquard equations with doubly critical exponents, Rocky Mountain J. Math., 49 (2019), 153-170.
doi: 10.1216/RMJ-2019-49-1-153. |
[17] |
G. D. Li and C. L. 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. |
[18] |
E. H. Lieb,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105.
doi: 10.1002/sapm197757293. |
[19] |
P.-L. Lions,
The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[20] |
L. Ma and L. Zhao,
Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.
doi: 10.1007/s00205-008-0208-3. |
[21] |
V. Moroz and J. Van Schaftingen,
Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct Anal., 265 (2013), 153-184.
doi: 10.1016/j.jfa.2013.04.007. |
[22] |
P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, RI, 1986.
doi: 10.1090/cbms/065. |
[23] |
J. Seok,
Nonlinear Choquard equations: Doubly critical case, Appl. Math. Lett., 76 (2018), 148-156.
doi: 10.1016/j.aml.2017.08.016. |
[24] |
M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-662-02624-3. |
[25] |
J. Van Schaftingen and J. Xia,
Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent, J. Math. Anal. Appl., 464 (2018), 1184-1202.
doi: 10.1016/j.jmaa.2018.04.047. |
[26] |
M. Willem, Minimax Theorems, Birkhäuser Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[27] |
M. Willem, Functional Analysis, Springer New York, 2013.
doi: 10.1007/978-1-4614-7004-5. |
[28] |
M. B. Yang, J. C. de Albuquerque, E. D. Silva and M. L. Silva,
On the critical cases of linearly coupled Choquard systems, Appl. Math. Lett., 91 (2019), 1-8.
doi: 10.1016/j.aml.2018.11.005. |
[29] |
Qi Zhang,
Existence, uniqueness and multiplicity of positive solutions for schrödinger-Poisson system with singularity, J. Math. Anal. Appl., 437 (2016), 160-180.
doi: 10.1016/j.jmaa.2015.12.061. |
[30] |
F. Zhang and M. Du,
Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well, J. Differ. Equations, 269 (2020), 10085-10106.
doi: 10.1016/j.jde.2020.07.013. |
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