Local and global properties of solutions of quasilinear Hamilton–Jacobi equations

We study some properties of the solutions of (E) −Δpu+|∇u|q=0−Δpu+|∇u|q=0 in a domain Ω⊂RNΩ⊂RN, mostly when p≥q>p−1p≥q>p−1. We give a universal a priori estimate of the gradient of the solutions with respect to the distance to the boundary. We give a full classification of the isolated singularities of the nonnegative solutions of (E), a partial classification of isolated singularities of the negative solutions. We prove a general removability result expressed in terms of some Bessel capacity of the removable set. We extend our estimates to equations on complete noncompact manifolds satisfying a lower bound estimate on the Ricci curvature, and derive some Liouville type theorems.