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

Abstract

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. We emphasize that it seems much difficult to use the variational framework developed in the existed literature to study the above problem.