  The weak and strong comparison principles, respectively, are investigated for
quasi-linear elliptic boundary value problems with the $p$-Laplacian in one
space dimension. We treat the degenerate case of $2 < p < \infty$ and allow
also for the nontrivial convection velocity $b\colon [-1,1]\to \mathbb{R}$ in
the underlying domain $(-1,1)$. We establish the weak comparison principle
under a rather general, natural sufficient condition on the convection
velocity, $b(x)$, and the reaction function, ${\varphi}(x,u)$. Furthermore, we
establish also the strong comparison principle under a number of various
additional hypotheses. In contrast, with these hypotheses being violated, we
construct also a few rather natural counterexamples to the strong comparison
principle and discuss their applications to an interesting classical problem of
fluid flow in porous medium, seepage flow of fluids in inclined bed. Our
methods are based on a mixture of classical and new techniques.
