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

Abstract

AbstractIn this paper, we focus on the existence of positive solutions to the following planar Schrödinger–Newton system with general critical exponential growth $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta {u}+u+\phi u =f(u)&{} \text{ in }\,\,\mathbb {R}^2, \\ \Delta {\phi }=u^2 &{} \text{ in }\,\, \mathbb {R}^2, \end{array} \right. \end{aligned}$$ - Δ u + u + ϕ u = f ( u ) in R 2 , Δ ϕ = u 2 in R 2 , where $$f\in C^1(\mathbb {R},\mathbb {R})$$ f ∈ C 1 ( R , R ) . We apply a variational approach developed in [36] to study the above problem in the Sobolev space $$H^1(\mathbb {R}^2)$$ 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 $$\frac{f(t)}{t^3}$$ 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.