The Choquard logarithmic equation involving a nonlinearity with exponential growth

TL;DR

This paper studies a nonlinear logarithmic Choquard equation with exponential growth nonlinearity in two dimensions, proving existence of solutions using variational methods and symmetry considerations.

Contribution

It establishes the existence of nontrivial and ground state solutions for the Choquard logarithmic equation with exponential critical growth, including symmetric cases.

Findings

Existence of a nontrivial solution at the mountain pass level

Existence of a nontrivial ground state solution

Results under symmetric group actions

Abstract

In the present work we are concerned with the Choquard Logarithmic equation $-\Delta u + au + \lambda (\ln|\cdot|\ast |u|^{2})u = f(u)$ in $\mathbb{R}^2$, for $ a>0 $, $ \lambda >0 $ and a nonlinearity $f$ with exponential critical growth. We prove the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution. Also, we provide these results under a symmetric setting, taking into account subgroups of $ O(2) $.