Comparison of solutions of nonlinear differential equations with loaded level sets

We extend well-known comparison results to a class of partial differential equations with a divergent principal part containing a weight coefficient that depends on the measure of a level set of solution. Let $\Omega \subset {\mathbb{R}}^m$ be an open set with finite volume. Let $g_0(x,u) = \Phi(\mathrm{meas\left\{ \chi \in \Omega \colon u(\chi) > u(x)\right\}})$, where $\Phi$ is a continuous nonnegative function. Let $u \colon \Omega \to [0, \infty)$ be a weak solution to
$$ - \sum_{j=1}^{m} \frac{\partial}{\partial x_j} \left( g(x,u) \cdot {|\nabla u|}^{p-2} \frac{\partial u}{\partial x_j}\right) = f(x) + k {|\nabla u|}^{q} $$
subject to homogeneous boundary conditions, where $g(x,u) \ge g_0(x,u), k\ge 0$ and $f \in L^1 (\Omega)$. We prove that under certain assumptions there is a weak nonnegative solution $V \colon {\Omega}^* \to [0, \infty)$ to homogeneous Dirichlet problem for
$$ - \sum_{j=1}^{m} \frac{\partial}{\partial x_j} \left( g_0(x,V) \cdot {|\nabla V|}^{p-2} \frac{\partial V}{\partial x_j}\right) = f(x) + k {|\nabla V|}^{q} $$
such that $u^* \le V$ and $\int\limits_{\Omega} {|\nabla u|}^{p} dx \le \int\limits_{{\Omega}^*} {|\nabla V|}^{p} dx$. Here $\Omega^*$ is the open ball whose volume coincides with the volume of $\Omega$ and $u^*$ is the Schwarz symmetrization of $u$.