Ground States for Planar Generalized Quasilinear Schrödinger Equation with Choquard Nonlinearity

In this paper, we consider the following generalized quasilinear Schrödinger equation \(\begin{aligned} -\text {div}(g^{2}(u)\nabla u)+g(u)g'(u)|\nabla u|^{2}+ V(x)u=\left( I_{\mu }*F(u)\right) f(u), \ x \in \mathbb {R}^{2}, \end{aligned}\) $$\begin{array}{c}\hfill -\text{div}(g^2(u)\mathrm{\nabla }u)+g(u)g^{\prime }(u){|\mathrm{\nabla }u|}^2+V(x)u=\left(I_{\mu },,\ast ,F,(u)\right)f(u),x\in {\mathbb{R}}^2,\end{array}$$ where V(x) is a 1-periodic function, \(I_{\mu }=\frac{1}{|x|^\mu }\) $I_{\mu }=\frac{1}{{|x|}^{\mu }}$ , \(\mu \in (0,2)\) $\mu \in (0,2)$ and F is the primitive of f. The main feature of this paper is the nonlinearity f satisfies the critical exponential growth with respect to Trudinger-Moser inequality. Under some appropriate assumptions on g and f, by using a change of variables, a version of Trudinger-Moser inequality and Hardy-Littlewood-Sobolev inequality, we obtain the existence of least energy solutions with subcritical exponential growth and Nehari-type ground state solutions with critical exponential growth via monotonicity condition instead of Ambrosetti-Rabinowitz condition. In particular, we introduce a general lower bound of \(\frac{tF(t)}{e^{\zeta _0t^{2\alpha }}}\) $\frac{tF(t)}{e^{{\zeta }_0{t}^{2\alpha }}}$ near infinity, and our results extend some ones discussed in Chen et al. [J. Geom. Anal. 33 (2023) 299].