Schr\"odinger-Newton equations in dimension two via a Pohozaev-Trudinger log-weighted inequality

TL;DR

This paper establishes the existence of finite energy solutions for a two-dimensional Schr"odinger-Newton type equation with exponential nonlinearity, using a novel weighted Pohozaev--Trudinger inequality to handle the unique challenges posed by the logarithmic kernel.

Contribution

It introduces a new weighted Pohozaev--Trudinger inequality tailored for the Schr"odinger-Newton equations in 2D, enabling the analysis of exponential growth nonlinearities.

Findings

Existence of variational solutions proven

Development of a new weighted inequality for the problem

Handling of exponential nonlinearity with logarithmic kernel

Abstract

We study the following Choquard type equation in the whole plane $(C) -\Delta u+V(x)u=(I_2\ast F(x,u))f(x,u),x\in\mathbb{R}^2$ where $I_2$ is the Newton logarithmic kernel, $V$ is a bounded Schr\"odinger potential and the nonlinearity $f(x,u)$, whose primitive in $u$ vanishing at zero is $F(x,u)$, exhibits the highest possible growth which is of exponential type. The competition between the logarithmic kernel and the exponential nonlinearity demands for new tools. A proper function space setting is provided by a new weighted version of the Pohozaev--Trudinger inequality which enables us to prove the existence of variational, in particular finite energy solutions to $(C)$.