No abstract

In a series of papers the question of uniqueness of radial ground states of the equation ∆u + f(u) = 0 and of various related equations has been studied. It is remarkable that throughout this work (except in very special circumstances) nowhere is a spatially dependent term taken into consideration. Here we shall make a first attempt to study the uniqueness of ground states for such spatially dependent equations and to establish qualitative properties of solutions for this purpose. Mathematics Subject Classification (2000). Primary 35J70, Secondary 35J60

We study some properties of the solutions of (E) −∆ p u + |∇u| q = 0 in a domain Ω ⊂ R N , mostly when p ≥ q > p − 1. We give a universal 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 non compact manifolds satisfying a lower bound estimate on the Ricci curvature, and derive some Liouville type theorems.

No abstract