Asymptotically linear Schrödinger–Poisson systems with potentials vanishing at infinity

In this paper, we are concerned with the following nonlinear Schrödinger–Poisson equationsequation(PP){−Δu+V(x)u+λϕ(x)u=K(x)f(u),x∈R3,−Δϕ=u2,lim|x|→∞ϕ(x)=0, where λ>0λ>0 is a parameter, the potential V(x)V(x) may be vanishing at infinity, f(s)f(s) is asymptotically linear at infinity, that is f(s)∼O(s)f(s)∼O(s) as s→∞s→∞. For this kind of potential, it seems difficult to find solutions in H1(R3)H1(R3). Under some assumptions on V(x),K(x)V(x),K(x) and f(s)f(s), we prove that problem (aaaa) has a positive solution for λ small and has no any nontrivial solution for λ large.