In this paper we will be concerned with the problem \(\matrix{{ - \Delta u - {1 \over 2}\Delta ({a( x ){u^2}})u + V( x )u = f(u),} & {x \in {\mathbb{R}^2}}},\) where V is a potential continuous and f: ℝ → ℝ is a superlinear continuous function with exponential subcritical or exponential critical growth. We use as a main tool the Nehari manifold method in order to show existence of nonnegative solutions and existence of nodal solutions. Our results complement the classical result of “Solutions for quasilinear Schrdinger equations via the Nehari method” due to Jia–Quan Liu, Ya–Qi Wang and Zhi-Qiang Wang in the sense that in this article we are considering nonlinearity of the exponential type.
