On the existence and multiplicity of solutions for the $ N $-Choquard logarithmic equation with exponential critical growth

TL;DR

This paper establishes the existence and multiplicity of solutions for a logarithmic Choquard equation with exponential critical growth, adapting techniques from fractional Laplacian problems to this nonlocal, nonlinear setting.

Contribution

It extends variational methods to the Choquard logarithmic equation with critical exponential growth, proving existence of solutions and infinitely many solutions in subcritical cases.

Findings

Existence of nontrivial solutions at mountain pass level

Existence of ground state solutions in critical case

Infinitely many solutions in subcritical case

Abstract

In the present work we briefly explain how to adapt techniques already used in fractional and $p$-fractional Laplacian cases to obtain the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution, for the critical case, and the existence of infinitely many solutions, for the subcritical case, to the Choquard Logarithmic equation, $-\Delta_N u + a(x)|u|^{N-2}u + \lambda (\ln|\cdot|\ast |u|^{N})|u|^{N-2}u = f(u) \textrm{ in } \mathbb{R}^N $, where $ a:\mathbb{R}^N \rightarrow \mathbb{R} $, $ \lambda >0 $, $ N \geq 3 $ and $f: \mathbb{R} \rightarrow [0, \infty) $ is continuous function that behaves like $ \exp(\alpha |u|^{\frac{N}{N-1}}) $ at infinity, for $ \alpha >0 $.