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August  2021, 29(3): 2475-2488. doi: 10.3934/era.2020125

The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations

Li Cai and  Fubao Zhang ,

School of Mathematics, Southeast University, Nanjing 210096, China

* Corresponding author: Fubao Zhang, supported by NNSFC 11671077

Received  August 2020 Revised  October 2020 Published  August 2021 Early access  December 2020

In this paper, we study the following Schrödinger-Poisson equations with double critical exponents:
$ \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*} $
where
$ \Omega $
 is a bounded domain in
$ \mathbb{R}^3 $
 with Lipschitz boundary,
$ \lambda $
 is a real parameter satisfying suitable conditions. Using variational methods, we show the existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson equations.
Keywords: Schrödinger-Poisson equation, Brezis-Nirenberg type, double critical exponents, nontrivial solutions, variational methods.
Mathematics Subject Classification: Primary: 35J15, 35J20; Secondary: 35J50.
Citation: Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, 2021, 29 (3) : 2475-2488. doi: 10.3934/era.2020125
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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. Seok, Nonlinear Choquard equations: Doubly critical case, Appl. Math. Lett., 76 (2018), 148-156.  doi: 10.1016/j.aml.2017.08.016.  Google Scholar

[24]

M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-662-02624-3.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

M. Willem, Functional Analysis, Springer New York, 2013. doi: 10.1007/978-1-4614-7004-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. Seok, Nonlinear Choquard equations: Doubly critical case, Appl. Math. Lett., 76 (2018), 148-156.  doi: 10.1016/j.aml.2017.08.016.  Google Scholar

[24]

M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-662-02624-3.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

M. Willem, Functional Analysis, Springer New York, 2013. doi: 10.1007/978-1-4614-7004-5.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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