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 $\mathbf{R}^n$ 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 $−Lu=f(x, u, ∇u)+ω$ where $f(x, u, ∇u)\asymp a(x)|∇u|^{q_1} +b(x)|u|^{q_2}$, and $L$ is a uniformly elliptic operator.