In this paper, we consider the following generalized quasilinear Schrödinger equation \(\begin{aligned} -\text {div}(g^{2}(u)\nabla u)+g(u)g'(u)|\nabla u|^{2}+ V(x)u=\lambda Q(x)|u|^{q-2}u+f(u), \ x \in {\mathbb {R}}^{2}, \end{aligned}\) where \(q\in (1,2)\) , \(\lambda >0\) , and Q is indefinite in sign, and f fulfills the critical exponential growth with respect to Trudinger-Moser inequality. Different from the usual polynomial growth condition as in [27, 45], this paper employs the asymptotic condition \(\liminf \limits _{t\rightarrow +\infty } \frac{tf(t)}{e^{\zeta _0 t^{2\alpha }}}\ge \kappa \) to restore the compactness due to the critical exponential growth. By applying variational methods and a change of variables and some techniques in [9] to estimate finely the minimax level of the energy functional, we obtain the existence of two different nontrivial solutions for this problem in the Sobolev space \(H^1({{\mathbb {R}}}^2)\) .
