Weighted Sobolev Spaces and an Eigenvalue Problem for an Elliptic Equation with $L^1$ Data

The aim of this work is to study the continuity and compactness of the operators $W^{1,q}(\mathrm{\Omega };{\mathtt{V}}_0,{\mathtt{V}}_1)\to L^{q_0}(\mathrm{\Omega };{\mathtt{V}}_2)$ and $W^{1,q}(\mathrm{\Omega };{\mathtt{V}}_0,{\mathtt{V}}_1)\to L^{q_1}(\mathrm{\partial }\mathrm{\Omega };\mathtt{W})$ in weighted Sobolev spaces. To study additional properties of these Sobolev spaces, we will also study the equation: $$\{\begin{array}{rlrl}{\displaystyle -\mathrm{div}\left({\mathtt{V}}_1\mathrm{\nabla }u\right)+{\mathtt{V}}_{0}u}& {\displaystyle =\lambda {\mathtt{V}}_2\tau u+{\mathtt{V}}_2{f}_0}& {\displaystyle }& {\displaystyle \text{ in }\mathrm{\Omega },}\\ {\displaystyle {\mathtt{V}}_1\frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }}& {\displaystyle ={\mathtt{W}}_1{f}_1}& {\displaystyle }& {\displaystyle \text{ on }\mathrm{\partial }\mathrm{\Omega },}\end{array}$$ where $\mathrm{\Omega }$ is an open subset of a Riemannian manifold, $\lambda$ is a real number, $f_0\in L^1(\mathrm{\Omega };{\mathtt{V}}_0),f_1\in L^1(\mathrm{\partial }\mathrm{\Omega };\mathtt{W})$, $\tau$ is a function that changes sign, and ${\mathtt{V}}_i,\mathtt{W},{\mathtt{W}}_1$ are weight functions satisfying suitable conditions. We aim to obtain existence results similar to those for the case where the data are given in $L^2(\mathrm{\Omega };{\mathtt{V}}_0)$ and $L^2(\mathrm{\partial }\mathrm{\Omega };\mathtt{W})$. For the case where $f_0=0$ and $f_1=0$, we are also interested in studying the limit ess ${\sup }_{\mathrm{\Omega }\mathrm{\backslash }{\mathrm{\Omega }}_m}\mathrm{\mid }u\mathrm{\mid }\to 0$, where ${\mathrm{\Omega }}_m$ is a sequence of open sets such that ${\mathrm{\Omega }}_m\subset {\mathrm{\Omega }}_{m+1}$.