Quasilinear Schr\"odinger equations with Stein-Weiss type convolution and critical exponential nonlinearity in $\mathbb R^N$

In this article, we investigate the existence of the positive solutions to the following class of quasilinear {Schr\"odinger} equations involving Stein-Weiss type convolution \begin{align*} -\Delta_N u -\Delta_N (u^{2})u +V(x)|u|^{N-2}u= \left(\int_{\mathbb R^N}\frac{F(y,u)}{|y|^\beta|x-y|^{\mu}}~dy\right)\frac{f(x,u)}{|x|^\beta} \;\; \text{ in}\; \mathbb R^N, \end{align*} where $N\geq 2,\,$ $0<\mu<N,\, \beta\geq 0,$ and $2\beta+\mu\leq N.$ The potential $V:\mathbb R^N\to \mathbb R$ is a continuous function satisfying $0<V_0\leq V(x)$ for all $x\in \mathbb R^N$ and some appropriate assumptions. The nonlinearity $f:\mathbb R^N\times \mathbb R\to \mathbb R$ is a continuous function with critical exponential growth in the sense of the Trudinger-Moser inequality and $F(x,s)=\int_{0}^s f(x,t)dt$ is the primitive of $f$.