Ground states for planar axially Schrödinger–Newton system with an exponential critical growth

In this paper, we study the following planar Schrödinger–Newton system: { − Δ u + V ( x ) u + λ ϕ u = f ( x , u ) in R 2 , Δ ϕ = u 2 in R 2 , where V , f are axially symmetric about x , V is positive, and f is super-linear at zero and exponential critical at infinity. Using a weaker condition [ f ( x , u ) u 3 − f ( x , t u ) ( t u ) 3 ] sign ( 1 − t ) + θ V ( x ) | 1 − t 2 | ( t u ) 2 ≥ 0 , ∀ x ∈ R 2 , t > 0 , u ≠ 0 with θ ∈ [ 0 , 1 ) instead of the Nehari type monotonic condition on f ( x , u ) | u | 3 , we obtain a ground state solution of the above problem via variational methods.