A strongly degenerate quasilinear elliptic equation

We prove existence and uniqueness of entropy solutions for the quasilinear elliptic equation u-diva(u,Du)=v, where 0⩽v∈L1(RN)∩L∞(RN), a(z,ξ)=∇ξf(z,ξ), and f is a convex function of ξ with linear growth as ∥ξ∥→∞, satisfying other additional assumptions. In particular, this class of equations includes the elliptic problems associated to a relativistic heat equation and a flux limited diffusion equation used in the theory of radiation hydrodynamics, respectively. In a second part of this work, using Crandall-Liggett's iteration scheme, this result will permit us to prove existence and uniqueness of entropy solutions for the corresponding parabolic Cauchy problem.