On Weighted Orlicz-Sobolev inequalities

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