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

Li Cai; Fubao Zhang; Li Cai; Fubao Zhang

doi:10.3934/era.2020125

Electronic Research Archive

2021, Volume 29, Issue 3: 2475-2488. doi: 10.3934/era.2020125

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

Li Cai ,
Fubao Zhang ^,

School of Mathematics, Southeast University, Nanjing 210096, China

Received: 01 August 2020 Revised: 01 October 2020 Published: 14 December 2020
Primary: 35J15, 35J20; Secondary: 35J50

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.
- Schrödinger-Poisson equation,
- Brezis-Nirenberg type,
- double critical exponents,
- nontrivial solutions,
- variational methods
Citation: Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations[J]. Electronic Research Archive, 2021, 29(3): 2475-2488. doi: 10.3934/era.2020125

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Abstract

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.

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]	Existence of semiclassical ground state solutions for a generalized Choquard equation. J. Differ. Equations. (2014) 257: 4133-4164.
[3]	Multiplicity and concentration of solutions for a quasilinear Choquard equation. J. Math. Phys. (2014) 55: 423-443.
[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]	On a system involving a critically growing nonlinearity. J. Math. Anal. Appl. (2012) 387: 433-438.
[6]	Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pur. Appl. Math. (1983) 36: 437-477.
[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]	An existence result for nonlinear elliptic problems involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire (1985) 2: 463-470.
[9]	Some existence results for superlinear elliptic boundary value problems involving critical exponents. J. Funct. Anal. (1986) 69: 289-306.
[10]	Multiple solutions to a magnetic nonlinear Choquard equation. Z. Angew. Math. Phys. (2012) 63: 233-248.
[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]	The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation. Sci. China Math. (2018) 61: 1219-1242.
[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]	Ground states for Choquard equations with doubly critical exponents. Rocky Mountain J. Math. (2019) 49: 153-170.
[17]	Existence of a ground state solution for Choquard equation with the upper critical exponent. Comput. Math. Appl. (2018) 76: 2635-2647.
[18]	Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Studies in Appl. Math. (1976/77) 57: 93-105.
[19]	The Choquard equation and related questions. Nonlinear Anal. (1980) 4: 1063-1072.
[20]	Classification of positive solitary solutions of the nonlinear Choquard equation. Arch. Ration. Mech. Anal. (2010) 195: 455-467.
[21]	Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics. J. Funct Anal. (2013) 265: 153-184.
[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]	Nonlinear Choquard equations: Doubly critical case. Appl. Math. Lett. (2018) 76: 148-156.
[24]	M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-662-02624-3
[25]	Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent. J. Math. Anal. Appl. (2018) 464: 1184-1202.
[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]	On the critical cases of linearly coupled Choquard systems. Appl. Math. Lett. (2019) 91: 1-8.
[29]	Existence, uniqueness and multiplicity of positive solutions for schrödinger-Poisson system with singularity. J. Math. Anal. Appl. (2016) 437: 160-180.
[30]	Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well. J. Differ. Equations (2020) 269: 10085-10106.

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