# American Institute of Mathematical Sciences

doi: 10.3934/era.2020125

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

 School of Mathematics, Southeast University, Nanjing 210096, China

* Corresponding author: Fubao Zhang, supported by NNSFC 11671077

Received  August 2020 Revised  October 2020 Published  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.
Citation: Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, doi: 10.3934/era.2020125
##### References:

show all references

##### References:
 [1] 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 [2] Maoding Zhen, Binlin Zhang, Xiumei Han. A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021115 [3] 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 [4] Fangyi Qin, Jun Wang, Jing Yang. Infinitely many positive solutions for Schrödinger-poisson systems with nonsymmetry potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021054 [5] 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 [6] Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021039 [7] Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2947-2969. doi: 10.3934/dcds.2020392 [8] Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447 [9] Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030 [10] Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3651-3682. doi: 10.3934/dcds.2021011 [11] Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3817-3836. doi: 10.3934/dcds.2021018 [12] Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216 [13] Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021047 [14] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [15] 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 [16] Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021 [17] Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028 [18] Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031 [19] Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $H^1$. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019 [20] Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267