  The paper is devoted to the study of positive solutions of a second-order
linear elliptic equation in divergence form in a domain $D\subseteq
\mathbb{R}^n$ that satisfy an oblique boundary condition on a portion of
$\partial D$. First, we study the degenerate mixed boundary value problem
  $$
  \begin{cases}
  Pu=f & \text{in } D,
  \\ Bu = 0 & \text{on } \partial D_{\mathrm{Rob}},
  \\ u=0& \text{on } \partial D_{\mathrm{Dir}},
  \end{cases}
  $$
  where $D$ is a bounded Lipschitz domain, $\partial D_{\mathrm{Rob}}$ is a
relatively open portion of $\partial D$, $\partial D_{\mathrm{Dir}}$ is a
closed set of $\partial D$, and $B$ is an oblique (Robin) boundary operator
defined on $\partial D_{\mathrm{Rob}}$. In particular, we discuss the unique
solvability of the above problem, the existence of a principal eigenvalue, and
the existence of a positive minimal Green function. Then we establish a
criticality theory for positive weak solutions of the operator $(P,B)$ in a
general domain with no boundary condition on $\partial D_{\mathrm{Dir}}$ and no
growth condition at infinity. The paper generalizes and extends results
obtained by Pinchover and Saadon (2002) for classical solutions of such a
problem, where stronger regularity assumptions on the coefficients of $(P,B)$,
and $\partial D_{\mathrm{Rob}}$.
