  This work is concerned with the existence and multiplicity of solutions for
the following class of quasilinear problems $$
-\Delta_{\Phi}u+\phi(|u|)u=f(u)~\text{in} ~\Omega_{\lambda}, u(x)>0
~\text{in}~\Omega_{\lambda}, u=0~ \mbox{on} ~\partial\Omega_{\lambda}, $$ where
$\Phi(t)=\int_0^{|t|} \phi(s) s \, ds $ is an $N-$function, $\Delta_{\Phi}$ is
the $\Phi-$Laplacian operator, \linebreak $\Omega_{\lambda}=\lambda \Omega,$
$\Omega$ is a smooth bounded domain in $\mathbb{R}^N,$ $N \geq 2$, $\lambda$ is
a positive parameter and $f: \mathbb{R}\rightarrow \mathbb{R}$ is a continuous
function. Here, we use variational methods to get multiplicity of solutions by
using of Lusternik-Schnirelmann category of ${\Omega}$ in itself.
