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

Abstract

In this paper, we are concerned with the following Schrödinger-Poisson system with critical nonlinearity and critical nonlocal term due to the Hardy-Littlewood-Sobolev inequality \begin{equation}\begin{cases} -Δu+u+λϕ|u|^3u =|u|^4u+ |u|^{q-2}u,\ \ &\ x \in \mathbb{R}^{3},\\[2mm] -Δϕ=|u|^5, \ \ &\ x \in \mathbb{R}^{3}, \end{cases} \end{equation} where $\lambda \in \mathbb{R}$ is a parameter and $q\in (2,6)$. If $\lambda \ge (\frac{q+2}{8}{)}^2$ and $q\in (2,6)$, the above system has no nontrivial solution. If $\lambda \in ({\lambda }^{\ast },0)$ for some ${\lambda }^{\ast }<0$, we obtain a least energy radial sign-changing solution $u_{\lambda }$ to the above system. Furthermore, we consider $\lambda$ as a parameter and analyze the asymptotic behavior of $u_{\lambda }$ as $\lambda \to 0^{-}$.