  In this paper we prove an existence result of multiple positive solutions for
the following quasilinear problem \begin{equation*} \left\{
\begin{array}[c]{ll} -\Delta u - \Delta (u^2)u = |u|^{p-2}u & \mbox{ in }
\Omega u= 0 &\mbox{ on } \partial\Omega, \end{array} \right. \end{equation*}
where $\Omega$ is a smooth and bounded domain in $\mathbb R^{N},N\geq3$. More
specifically we prove that, for $p$ near the critical exponent
$22^{*}=4N/(N-2)$, the number of positive solutions is estimated below by
topological invariants of the domain $\Omega$: the Ljusternick-Schnirelmann
category and the Poincar\'e polynomial.
