Existence and multiplicity of solutions for a class of quasilinear problems in Orlicz-Sobolev spaces

This work is concerned with the existence and multiplicity of solutions for the following class of quasilinear problems $$-{\mathrm{\Delta }}_{\mathrm{\Phi }}u+\varphi (\mathrm{\mid }u\mathrm{\mid })u=f(u)\text{ in }{\mathrm{\Omega }}_{\lambda },u(x)>0\text{ in }{\mathrm{\Omega }}_{\lambda },u=0on\mathrm{\partial }{\mathrm{\Omega }}_{\lambda },$$ where $\mathrm{\Phi }(t)={\int }_0^{\mathrm{\mid }t\mathrm{\mid }}\varphi (s)sds$ is an $N-$function, ${\mathrm{\Delta }}_{\mathrm{\Phi }}$ is the $\mathrm{\Phi }-$Laplacian operator, \linebreak ${\mathrm{\Omega }}_{\lambda }=\lambda \mathrm{\Omega },$ $\mathrm{\Omega }$ is a smooth bounded domain in ${\mathbb{R}}^N,$ $N\ge 2$, $\lambda$ is a positive parameter and $f:\mathbb{R}\to \mathbb{R}$ is a continuous function. Here, we use variational methods to get multiplicity of solutions by using of Lusternik-Schnirelmann category of $\mathrm{\Omega }$ in itself.