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

Full-text: Open access


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.


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

Article information

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

1999 © Institut Mittag-Leffler


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.

Export citation


  • [AH] Adams, D. R. and Hedberg, L. I., Function Spaces and Potential Theory, Springer-Verlag, Berlin-Heidelberg, 1996.
  • [AP] Adams, D. R. and Pierre, M., Capacitary strong type estimates in semilinear problems, Ann. Inst. Fourier (Grenoble) 41:1 (1991), 117–135.
  • [Ag] Agmon, S., On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, in Methods of Functional Analysis and Theory of Elliptic Equations (Greco, D., ed.), pp. 19–52, Liguori, Naples, 1983.
  • [A1] Ancona, A., On strong barriers and an inequality of Hardy for domains in Rn, J. London Math. Soc. 34 (1986), 274–290.
  • [A2] Ancona, A., First eigenvalues and comparison of Green's functions for elliptic operators on manifolds or domains, J. Anal. Math. 72 (1997), 45–92.
  • [B] Baras, P., Semilinear problem with convex nonlinearity, in Recent Advances in Nonlinear Elliptic and Parabolic Problems (Bénilan, P., Chipot, M., Evans, L. C. and Pierre, M., eds.), Pitman Research Notes in Math. Sciences 208, pp. 202–215, Longman, Harlow, 1989.
  • [BP] Baras, P. and Pierre, M., Singularités éliminables pour des équations semilinéaires, Ann. Inst. Fourier (Grenoble) 34:1 (1984), 185–206.
  • [CZ] Chung, K. L. and Zhao, Z., From Brownian Motion to Schrödinger's Equation, Springer-Verlag, Berlin, 1995.
  • [GW] Grüter, M. and Widman, K.-O., The Green function for uniformly elliptic equations, Manuscripta Math. 37 (1982), 303–342.
  • [H] Hansson, K., Imbedding theorems of Sobolev type in potential theory, Math. Scand. 45 (1979), 77–102.
  • [Ha] Hartman, P., Ordinary Differential Equations, Republ. 2nd ed., Birkhäuser, Boston, Mass., 1982.
  • [HK] Hayman, W. K. and Kennedy, P. B., Subharmonic Functions, Vol. I, Academic Press, London-New York-San Francisco, 1976.
  • [He] Hedberg, L. I., On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505–510.
  • [HS] Hueber, H. and Sieveking, M., Uniform bounds for quotients of Green functions on C1,1-domains, Ann. Inst. Fourier (Grenoble) 32:1 (1982), 105–117.
  • [KV] Kalton, N. J. and Verbitsky, I. E., Nonlinear equations and weighted norm inequalities, to appear in Trans. Amer, Math. Soc.
  • [L] Lions, P. L., On the existence of positive solutions of semilinear elliptic equations, SIAM Rev. 24 (1982), 441–467.
  • [M1] Maz'ya, V. G., On the theory of the n-dimensional Schrödinger operator, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1145–1172 (Russian).
  • [M2] Maz'ya, V. G., Sobolev Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1985.
  • [MV] Maz'ya, V. G. and Verbitsky, I. E., Capacitary estimates for fractional integrals, with applications to partial differential equations and Sobolev multipliers, Ark. Mat. 33 (1995), 81–115.
  • [N] Nyström, K., Integrability of Green potentials in fractal domains, Ark. Mat. 34 (1996), 335–381.
  • [S] Schechter, M., Hamiltonians for singular potentials, Indiana Univ. Math. J. 22 (1972), 483–503.
  • [Si] Simon, B., Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447–526.
  • [St] Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N. J., 1970.
  • [VW] Verbitsky, I. E. and Wheeden, R. L., Weighted inequalities for fractional integrals and applications to semilinear equations, J. Funct. Anal. 129 (1995), 221–241.
  • [W] Widman, K.-O., Inequalities for the Green function and boundary continuity of the gradients of solutions of elliptic differential equations, Math. Scand. 21 (1967), 13–67.
  • [Z] Zhao, Z., Green function for Schrödinger operator and conditioned Feynman-Kac gauge, J. Math. Anal. Appl. 116 (1986), 309–334.