Ground states for Schrödinger-Poisson system with zero mass and the Coulomb critical exponent

Abstract This article focuses on the study of the following Schrödinger-Poisson system with zero mass: − Δ u + ϕ u = ∣ u ∣ u + f ( u ) , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 , \left\{\begin{array}{ll}-\Delta u+\phi u=| u| u+f\left(u),& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi ={u}^{2},& x\in {{\mathbb{R}}}^{3},\end{array}\right. where f f is a continuous function satisfying some general growth conditions, and it requires only to be super-quadratic growth at infinity and includes, in particular, the pure power function ∣ u ∣ p − 2 u {| u| }^{p-2}u with p ∈ ( 3 , 6 ) p\in \left(3,6) . The nonlinear term ∣ u ∣ u | u| u is the so-called Coulomb critical nonlinearity because it presents a certain scaling invariance and the mountain-pass geometry cannot be established when f = 0 f=0 . Few results are known to such case. By developing some delicate analyses and using detailed estimates, we obtain the existence of ground-states and least energy solution for the aforementioned system under some natural assumptions on f f .