Positive solutions of asymptotically linear Schrödinger–Poisson systems with a radial potential vanishing at infinity

In this paper, we study the Schrödinger–Poisson system equation(SP){−Δu+V(x)u+λϕ(x)u=K(x)f(u),inR3,−Δϕ=u2,u>0,inR3, and prove the existence of positive solutions for system (SP) when the nonlinearity ff has growth at most linear for λλ small, allowing the potential V(x)V(x) to vanish at infinity. In addition, also we obtain the nonexistence of a nontrivial positive solution for λ≥14.