  In this paper, we consider existence of positive solutions for the
Schr\"odinger quasilinear elliptic problem $$
  \left\{ \begin{array}{l} \Delta_pu+\Delta_p(|u|^{2\gamma})|u|^{2\gamma-2}u =
a(x)g(u)~ \mbox{on}~ \mathbb{R}^N,\\ u>0\ \mbox{in}~\mathbb{R}^N,\
u(x)\stackrel{\left|x\right|\rightarrow \infty}{\longrightarrow} \infty,
\end{array} \right. $$ where $a(x), ~x\in \mathbb{R}^N$ and $g(s)~s>0$ are a
nonnegative and continuous functions with $g$ being nonincreasing as well,
$\gamma>{1}/{2}$, and $N \geq 1$. By a dual approach we establish sufficient
conditions for existence and multiplicity of solutions for this problem.
