Existence of Semiclassical Ground State Solutions for a Class of N-Laplace Choquard Equation with Critical Exponential Growth

In this paper, we discuss the following N-Laplace Choquard equation \(\begin{aligned} \left\{ \begin{aligned}&- \varepsilon ^{N}\Delta _{N} v+ V(x)|v|^{N-2}v= \varepsilon ^{\mu -N}(|x|^{-\mu }*F(v))f( v),~~~~~ x\in \mathbb {R}^{N},\\&u\in W^{1,N}(\mathbb {R}^{N}), \end{aligned}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}& -{\epsilon }^N{\mathrm{\Delta }}_{N}v+{V(x)|v|}^{N-2}v={\epsilon }^{\mu -N}{(|x|}^{-\mu }\ast F(v))f(v),x\in {\mathbb{R}}^N,\hfill \\ \hfill & u\in W^{1,N}({\mathbb{R}}^N),\hfill \end{array}\right)\end{array}$$ where \(N\ge 2\) $N\ge 2$ , \(\mu \in (0,N)\) $\mu \in (0,N)$ , \(\Delta _{N}v= \text {div}(|\nabla v|^{N-2}\nabla v)\) ${\mathrm{\Delta }}_{N}v={\text{div}(|\mathrm{\nabla }v|}^{N-2}\mathrm{\nabla }v)$ is the N-Laplace operator, \(\varepsilon \) $\epsilon$ is a positive parameter, V is a differentiable potential, F is the primitive of f with critical exponential growth in the sense of Trudinger–Moser. To address the challenges stemming from the presence of an exponential growth given by f and the quasilinear nature of the equation, we conduct meticulous analyses. This enables us to establish an intricate threshold for the Mountain-Pass minimax level and demonstrate the presence and concentration of semiclassical ground state solutions for the equation in question.