Criteria of solvability for multidimensional Riccati equations

Abstract

We study the solvability problem for the multidimensional -Riccati equation - Δu=|∇u|q+ω, where q>1 and ω is an arbitrary nonnegative function (or measure). We also discuss connections with the classical problem of the existence of positive solutions for the Schrödinger equation - Δu-ωu=0 with nonnegative potential ω. We establish explicit criteria for the existence of global solutions on Rn in terms involving geometric (capacity) estimates or pointwise behavior of Riesz potentials, together with sharp pointwise estimates of solutions and their gradients. We also consider the corresponding nonlinear Dirichlet problem on a bounded domain, as well as more general equations of the type - Lu=f(cursive Greek chi, u, ∇u)+ω where f(cursive Greek chi, u, ∇u)(Equivalent to)a(cursive Greek chi)|∇u|q1+b(cursive Greek chi)\u|q2 , and L is a uniformly elliptic operator.