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 Ω ⊂ R𝑚 be an open set with finite volume. Let 𝑔0(𝑥, 𝑢) = Φ(︀meas {𝜒 ∈ Ω : 𝑢(𝜒) > 𝑢(𝑥)})︀, where Φ is a continuous nonnegative function. Let : Ω → [0,∞) be a weak solution to - Σ︁𝑗=1 𝜕𝑥𝑗 (︁𝑔(𝑥, 𝑢) · |∇𝑢|𝑝-2 𝜕𝑥𝑗 )︁= 𝑓(𝑥) + 𝑘|∇𝑢|𝑞, subject to homogeneous boundary conditions, where 𝑔(𝑥, 𝑢) ≥ 𝑔0(𝑥, 𝑢), ≥ 0 and ∈ 𝐿1(Ω). We prove that under certain assumptions there is a weak nonnegative solution : Ω⋆ → [0,∞) to homogeneous Dirichlet problem for - Σ︁𝑗=1 𝜕𝑥𝑗 (︁𝑔0(𝑥, ) · |∇𝑉 |𝑝-2 𝜕𝑥𝑗 )︁= 𝑓(𝑥) + 𝑘|∇𝑉 |𝑞, such that 𝑢⋆ 6 and rΩ |∇𝑢|𝑝𝑑𝑥 6 rΩ⋆ |∇𝑉 |𝑝𝑑𝑥. Here Ω⋆ is the open ball whose volume coincides with the volume of Ω and 𝑢⋆ is the Schwarz symmetrization of 𝑢.