### Electronic Research Archive

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

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

• 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.

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

### Related Papers:

• 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.

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沈阳化工大学材料科学与工程学院 沈阳 110142