Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential

In this paper, we study the existence and nonexistence of nontrivial solutions for the following Schrödinger-Poisson system with zero mass potential { − △ u + ϕu = − a | u | p − 2 u + f ( u ) , x ∈ R 3 , − △ ϕ = u 2 , x ∈ R 3 , where a> 0 and p ∈ ( 2 , 12 5 ) . We obtain the existence of nontrivial radial solutions and the nonexistence of nontrivial solutions for the above system under weaker assumptions on a and f . In particular, applying our results to the following system: { − △ u + ϕu = − | u | p − 2 u + | u | q − 2 u , x ∈ R 3 , − △ ϕ = u 2 , x ∈ R 3 , a sufficient and necessary condition is obtained on the existence of nontrivial radial solutions.