  Using minimax methods and Lusternik-Schnirelmann theory, we study multiple
positive solutions for the Schr\"{o}dinger - Kirchhoff equation $$
M\left(\dis\int_{\Omega_{\lambda}}|\nabla
u|^{2}dx+\dis\int_{\Omega_{\lambda}}u^{2}dx\right)\left[-\Delta u + u \right]=
f(u) $$ in $\Omega_{\lambda} = \lambda\Omega$. The set $\Omega \subset
\mathbb{R}^3$ is a smooth bounded domain, $\lambda>0$ is a parameter, $M$ is a
general continuous function and $f$ is a superlinear continuous function with
subcritical growth. Our main result relates, for large values of $\lambda$, the
number of solutions with the least number of closed and contractible in
$\bar{\Omega}$ which cover $\bar{\Omega}$.
