  We propose a new mathematical model of groundwater flow in porous medium
layered over inclined impermeable bed. In its full generality, this is a
free-surface problem. To obtain analytically tractable model, we use
generalized Dupuit-Forchheimer assumption for inclined impermeable bed. In this
way, we arrive at parabolic partial differential equation which is a
generalization of the classical Boussinesq equation. Novelty of our approach
consists in considering nonlinear constitutive law of the power type. Thus
introducing $p$-Laplacian-like differential operator into the Boussinesq
equation. Unlike in the classical case of the Boussinesq equation, the
convective term cannot be set aside from the main part of the diffusive term
and remains incorporated within it.
  In the sequel of the paper, we analyze qualitative properties of the
stationary solutions of our model. In particular, we study existence and
regularity of weak solutions for the following boundary value problem
\begin{equation*} \begin{aligned}
  &
  -
  \frac{\rm d}{{\rm d} x}
  \left[
  (u(x) + H) \left|\frac{{\rm d} u}{{\rm d} x}(x) \cos(\varphi) + \sin(\varphi)
\right|^{p - 2}
  \left(\frac{{\rm d} u}{{\rm d} x}(x) \cos(\varphi) + \sin(\varphi)\right)
  \right]
  &
  \begin{aligned}
  &
  =
  f(x)\,, & \qquad\qquad x \in (-1,1)\,,
  &
  u(-1) = u(1) = 0\,,&
  \end{aligned} \end{aligned} \end{equation*} where $p>1$, $H>0$, $\varphi\in
(0, \pi/2)$, $f\geq 0$, $f\in L^{1}(-1,1)$. In the case of $p>2$, we study
validity of Weak and Strong Maximum Principles as well. We use methods based on
the linearization of the $p$-Laplacian-type problems in the vicinity of known
solution, error estimates, and analysis of Green's function of the linearized
problem.
