On Weighted Orlicz-Sobolev inequalities

Let $\mathrm{\Omega }$ be an open subset of ${\mathbb{R}}^N$ with $N\ge 2.$ We identify various classes of Young functions $\mathrm{\Phi }$ and $\mathrm{\Psi }$, and function spaces for a weight function $g$ so that the following weighted Orlicz-Sobolev inequality holds: \begin{equation*}\label{ineq:Orlicz} Ψ^{-1}\left(\int_Ω|g(x)|,Ψ(|u(x)| )dx \right)\leq CΦ^{-1}\left(\int_ΩΦ(|\nabla u(x)|) dx \right),;;;\forall,u\in $\mathcal{C}$^1_c(Ω), \end{equation*} for some $C>0$. As an application, we study the existence of eigenvalues for certain nonlinear weighted eigenvalue problems.