Nontrivial Solutions for Schrödinger–Poisson Systems With Doubly Critical Exponents

Abstract

This article investigates the existence and nonexistence of nontrivial solutions for a Schrödinger–Poisson system with doubly critical exponents: $$\begin{array}{r}\{\begin{array}{ll}-\mathrm{\Delta }u+u+\lambda \varphi |u{|}^{3}u=|u{|}^{4}u+\mu |u{|}^{q-2}u,& \text{in}{\mathbb{R}}^3,\\ -\mathrm{\Delta }\varphi =|u{|}^5,& \text{in}{\mathbb{R}}^3,\end{array}\end{array}$$ where $\lambda >0$ , $\mu >0$ , and $q\in (2,6)$ . Under optimal parameter ranges, we prove both the existence of radial ground state solutions and the nonexistence of nontrivial solutions. This work completely resolves an open problem proposed in [X. Chen, C. Tang, Communications on Pure and Applied Analysis 23 (2024), 1011–1043].