Multiplicity of positive solutions for a quasilinear Schr\"odinger equation with an almost critical nonlinearity

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 $\mathrm{\Omega }$ is a smooth and bounded domain in ${\mathbb{R}}^N,N\ge 3$. More specifically we prove that, for $p$ near the critical exponent ${22}^{\ast }=4N\mathrm{/}(N-2)$, the number of positive solutions is estimated below by topological invariants of the domain $\mathrm{\Omega }$: the Ljusternick-Schnirelmann category and the Poincar'e polynomial.