Existence of ground state solutions for the Schrödinger–Poisson systems

In this paper, we consider the following Schrödinger–Poisson system-ε2▵u+V(x)u+K(x)ϕu=a(x)|u|p-1u,inR3,-ε2▵ϕ=K(x)u2,inR3with p∈(1,5)p∈(1,5) and ε>0ε>0. We not only investigate the existence of ground state solutions but also employ bifurcation theory to show that the norm of the given ground state solutions tends to zero as the Planck constant ε→0+ε→0+. Moreover, we also study some non-existence results.