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

TL;DR

This paper investigates the existence, nonexistence, and asymptotic behavior of least energy sign-changing solutions for a Schrödinger-Poisson system with critical nonlinearity and nonlocal terms in \\mathbb{R}^3.

Contribution

It establishes conditions for existence and nonexistence of solutions and analyzes their asymptotic behavior as a parameter approaches zero.

Findings

No solutions when \lambda \ge (\frac{q+2}{8})^2.

Existence of least energy radial sign-changing solutions for \lambda in (\lambda^*, 0).

As \lambda \to 0^-, solutions exhibit specific asymptotic behavior.

Abstract

In this paper, we are concerned with the following Schr\"{o}dinger-Poisson system with critical nonlinearity and 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},\\[2mm] -\Delta \phi=|u|^5, \ \ &\ x \in \mathbb{R}^{3}, \end{cases} \end{equation} where $\lambda\in \mathbb{R}$ is a parameter and $q\in(2,6)$. If $\lambda\ge (\frac{q+2}{8})^2$ and $q\in(2,6)$, the above system has no nontrivial solution. If $\lambda\in (\lambda^*,0)$ for some $\lambda^*<0$, 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 $\lambda\to 0^-$.