  We consider the equation $- \Delta u+V(x)u- k(\Del(|u|^{2}))u=g(x,u), u>0, x
\in {\BR}^2,$ where $V:{\BR}^2\to {\BR}$ and $g:{\BR}^2 \times {\BR}\to {\BR}$
are two continuous $1-$periodic functions. Also, we assume $g$ behaves like
$\exp (\beta |u|^4)$ as $|u|\to \infty.$ We prove the existence of at least one
weak solution $u \in H^1({\BR}^2)$ with $u^2 \in H^1({\BR}^2).$ Mountain pass
in a suitable Orlicz space together with Moser-Trudinger are employed to
establish this result. Such equations arise when one seeks for standing wave
solutions for the corresponding quasilinear Schr\"{o}dinger equations.
Schr\"{o}dinger equations of this type have been studied as models of several
physical phenomena. The nonlinearity here corresponds to the superfluid film
equation in plasma physics.

Analysis of PDEs