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

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 $.