Quasilinear logarithmic Choquard equations with exponential growth in $\mathbb{R}^N$

TL;DR

This paper studies a nonlocal Choquard equation with exponential growth in the critical Sobolev space, introducing a new inequality to establish the existence of finite energy solutions.

Contribution

It develops a novel log-weighted Pohozaev-Trudinger inequality to handle the exponential nonlinearity in a nonlocal Choquard equation at the critical Sobolev level.

Findings

Established a new functional framework using the log-weighted inequality.

Proved existence of finite energy solutions for the nonlocal Choquard equation.

Extended the analysis to the critical exponential growth case in Sobolev space.

Abstract

We consider the $N$-Laplacian Schr\"odinger equation strongly coupled with higher order fractional Poisson's equations. When the order of the Riesz potential $\alpha$ is equal to the Euclidean dimension $N$, and thus it is a logarithm, the system turns out to be equivalent to a nonlocal Choquard type equation. On the one hand, the natural function space setting in which the Schr\"odinger energy is well defined is the Sobolev limiting space $W^{1,N}(\mathbb{R}^N)$, where the maximal nonlinear growth is of exponential type. On the other hand, in order to have the nonlocal energy well defined and prove the existence of finite energy solutions, we introduce a suitable $log$-weighted variant of the Pohozaev-Trudinger inequality which provides a proper functional framework where we use variational methods.