Nontrivial Solutions for Schrödinger–Poisson Systems With Doubly Critical Exponents

This article investigates the existence and nonexistence of nontrivial solutions for a Schrödinger–Poisson system with doubly critical exponents: { − Δ u + u + λ ϕ | u | 3 u = | u | 4 u + μ | u | q − 2 u , in R 3 , − Δ ϕ = | u | 5 , in R 3 , where λ > 0 , μ > 0 , and q ∈ ( 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].