# American Institute of Mathematical Sciences

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

 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.
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
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