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

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