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

Abstract

This work is concerned with the existence and multiplicity of solutions for the following class of quasilinear problems $$ -Δ_Φu+ϕ(|u|)u=f(u)~\text{in} ~Ω_λ, u(x)>0 ~\text{in}~Ω_λ, u=0~ \mbox{on} ~\partialΩ_λ, $$ where $\Phi (t)={\int }_0^{|t|}\varphi (s)sds$ is an $N-$function, ${\Delta }_{\Phi }$ is the $\Phi -$Laplacian operator, \linebreak ${\Omega }_{\lambda }=\lambda \Omega ,$ $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N,$ $N\ge 2$, $\lambda$ 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 $\Omega$ in itself.