Ground state solutions for the nonlinear Schrödinger‐Poisson systems with sum of periodic and vanishing potentials

We study the existence of ground state solutions for the following Schrödinger‐Poisson equations:− Δ u + V ( x ) u + ϕ u = f ( x , u ) ,inR 3 ,− Δ ϕ = u 2 , inR 3 ,where V = V p + V l o c ∈ L ∞ ( R 3 ) is the sum of a periodic potential V p and a localized potential V l o c and f satisfies the subcritical or critical growth. Although the Nehari‐type monotonicity assumption on f is not satisfied in the subcritical case, we obtain the existence of a ground state solution as a minimizer of the energy functional on Nehari manifold. Moreover, we show that the existence and nonexistence of ground state solutions are dependent on the sign of V l o c .