  Using variational methods combined with perturbation arguments, we study the
existence of nontrivial classical solution for the quasilinear Schr\"{o}dinger
equation
  \begin{equation*}\label{1.1}
  -\Delta u+ V(x)u+ \frac{\kappa}{2}[\Delta |u|^2]u=l(u),\ x\in\mathbb{R}^N,
  \end{equation*} where $V:\mathbb{R}^N\rightarrow \mathbb{R}$ and
$l:\mathbb{R} \to \mathbb{R}$ are continuous function, $\kappa$ is a parameter
and $N\geq 3$. This model has been proposed in plasma physics and nonlinear
optics. As a main novelty with respect to some previous results, we are able to
deal with the case $\kappa>0$.
