  We prove that weakly differentiable weights $w$ which, together with their
reciprocals, satisfy certain local integrability conditions, admit a unique
associated first-order $p$-Sobolev space, that is \[H^{1,p}(\mathbb{R}^d,w\,\d
x)=V^{1,p}(\mathbb{R}^d,w\,\d x)=W^{1,p}(\mathbb{R}^d,w\,\d x),\] where
$d\in\N$ and $p\in [1,\infty)$. If $w$ admits a (weak) logarithmic gradient
$\nabla w/w$ which is in $L^q_{\text{loc}}(w\,\d x;\R^d)$, $q=p/(p-1)$, we
propose an alternative definition of the weighted $p$-Sobolev space based on an
integration by parts formula involving $\nabla w/w$. We prove that weights of
the form $\exp(-\beta |\cdot|^q-W-V)$ are $p$-admissible, in particular,
satisfy a Poincar\'e inequality, where $\beta\in (0,\infty)$, $W$, $V$ are
convex and bounded below such that $|\nabla W|$ satisfies a growth condition
(depending on $\beta$ and $q$) and $V$ is bounded. We apply the uniqueness
result to weights of this type. The associated nonlinear degenerate evolution
equation is also discussed.

Functional Analysis;Analysis of PDEs