An asymptotically linear Schrödinger–Poisson system on R3R3

In this paper, we are concerned with the following nonlinear Schrödinger–Poisson equation {−△u+βu+λϕu=f(x,u)u,x∈R3,−△ϕ=u2,lim|x|→∞ϕ(x)=0, where u,ϕ:R3→Ru,ϕ:R3→R are positive functions, the parameters β,λ>0β,λ>0, f(x,s)f(x,s) tend to p(x)p(x) and q(x)∈L∞(R3)q(x)∈L∞(R3), respectively, as s→0s→0 and s→+∞s→+∞. Under some simple assumptions onf(x,s)f(x,s), we give existence and nonexistence results, depending on the parameters ββand λλ.