Arkiv för Matematik

  • Ark. Mat.
  • Volume 37, Number 1 (1999), 87-120.

Criteria of solvability for multidimensional Riccati equations

Kurt Hansson, Vladimir G. Maz'ya, and Igor E. Verbitsky

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Abstract

We study the solvability problem for the multidimensional Riccati equation −∇u=|∇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(x, u, ∇u)+ω where

[math not provided]

, and L is a uniformly elliptic operator.

Note

Partially supported by the NSF and University of Missouri Research Board grants.

Article information

Source
Ark. Mat., Volume 37, Number 1 (1999), 87-120.

Dates
Received: 5 June 1997
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898618

Digital Object Identifier
doi:10.1007/BF02384829

Mathematical Reviews number (MathSciNet)
MR1673427

Zentralblatt MATH identifier
1087.35513

Rights
1999 © Institut Mittag-Leffler

Citation

Hansson, Kurt; Maz'ya, Vladimir G.; Verbitsky, Igor E. Criteria of solvability for multidimensional Riccati equations. Ark. Mat. 37 (1999), no. 1, 87--120. doi:10.1007/BF02384829. https://projecteuclid.org/euclid.afm/1485898618


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