A planar Schrödinger–Newton system with Trudinger–Moser critical growth

In this paper, we focus on the existence of positive solutions to the following planar Schrödinger–Newton system with general critical exponential growth aligned \ arrayll -Δ u+u+φ u =f(u)& in \,\,R², \\ Δ φ =u² & in \,\, R², array . aligned - Δ u + u + ϕ u = f ( u ) in R 2 , Δ ϕ = u 2 in R 2 , where f∈ C¹(R,R) f ∈ C 1 ( R , R ) . We apply a variational approach developed in [36] to study the above problem in the Sobolev space H¹(R²) H 1 ( R 2 ) . The analysis developed in this paper also allows to investigate the relation between a Riesz-type of Schrödinger–Newton systems and a logarithmic-type of Schrödinger–Poisson systems. Furthermore, this approach can overcome some difficulties resulting from either the nonlocal term with sign-changing and unbounded logarithmic integral kernel, or the critical nonlinearity, or the lack of monotonicity of f(t)/t³ f ( t ) t 3 . We emphasize that it seems much difficult to use the variational framework developed in the existed literature to study the above problem.