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.

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