Existence and asymptotic behavior of least energy sign-changing solutions for Schrödinger-Poisson systems with doubly critical exponents.

In this paper, we are concerned with the following Schrödinger-Poisson system with a critical nonlinearity and a critical nonlocal term due to the Hardy-Littlewood-Sobolev inequality$ \begin{equation*} \begin{cases} -\Delta u +u +\lambda\phi|u|^3u = |u|^4u +|u|^{q-2}u, \ \ &\ x\in\mathbb{R}^3,\\ -\Delta\phi = |u|^5, \ \ &\ x\in\mathbb{R}^3, \end{cases} \end{equation*} $where $ \lambda\in \mathbb{R} $ is a parameter and $ q\in(2,6) $. By employing variational methods, we present the existence and nonexistence results (depending on the parameters $ \lambda $ and $ q $) as follows:($ i $) if $ \lambda\ge (\frac{q+2}{8})^2 $ and $ q\in(2,6) $, there is no nontrivial solution of the above system;($ ii $) if $ \lambda\in (-\lambda^*,0] $ for some $ \lambda^*>0 $, we obtain a least energy radial sign-changing solution $ u_\lambda $ of the above system;($ iii $) we consider $ \lambda $ as a parameter and analyze the asymptotic behavior of $ u_\lambda $ as $ \lambda\to 0^- $.Moreover, a Schrödinger-Poisson system with a critical nonlocal term differs completely from a system with a subcritical nonlocal term. [ABSTRACT FROM AUTHOR]