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 $$ -Δ_Φu+ϕ(|u|)u=f(u)~\text{in} ~Ω_λ, u(x)>0 ~\text{in}~Ω_λ, u=0~ \mbox{on} ~\partialΩ_λ, $$ where $Φ(t)=\int_0^{|t|} ϕ(s) s \, ds$ is an $N-$function, $Δ_Φ$ is the $Φ-$Laplacian operator, \linebreak $Ω_λ=λΩ,$ $Ω$ is a smooth bounded domain in $\mathbb{R}^N,$ $N \geq 2$, $λ$ 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 $Ω$ in itself.

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