Ground state solutions for asymptotically periodic fractional Schrödinger–Poisson problems with asymptotically cubic or super‐cubic nonlinearities

In this paper, we consider the following fractional Schrödinger–Poisson problem:( − △ ) s u + V ( x ) u + K ( x ) ϕ ( x ) u = f ( x , u ) , x ∈ R 3 ,( − △ ) t ϕ = K ( x ) u 2 , x ∈ R 3 ,where s , t ∈(0,1],4 s +2 t >3, V ( x ), K ( x ), and f ( x , u ) are periodic or asymptotically periodic in x . We use the non‐Nehari manifold approach to establish the existence of the Nehari‐type ground state solutions in two cases: the periodic one and the asymptotically periodic case, by introducing weaker conditionslim | τ | → ∞∫ 0 τ f ( x , ξ ) d ξ | τ | σ= ∞ uniformly in x ∈ R 3with σ : = max { 3 , 4 − 2 t } andf ( x , τ )τ 3− f ( x , k τ )( k τ ) 3sign ( 1 − k ) + θ 0 V ( x ) | 1 − k 2 |( k τ ) 2⩾ 0 , ∀ x ∈ R 3 , k > 0 , τ ≠ 0with constant θ 0 ∈(0,1), instead oflim | τ | → ∞∫ 0 τ f ( x , ξ ) d ξ | τ | 4= ∞ uniformly in x ∈ R 3and the usual Nehari‐type monotonic condition on f ( x , τ )/| τ | 3 . Our results unify both asymptotically cubic or super‐cubic nonlinearities, which are new even for s = t =1. Copyright © 2017 John Wiley & Sons, Ltd.