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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