zbMATH — the first resource for mathematics

Criteria of solvability for multidimensional Riccati equations. (English) Zbl 1087.35513
The authors study the solvability problem for the generalized Riccati equation \[ -\Delta u=| \nabla u| ^q+\omega\tag{1.1} \] where \(q>1\) and \(\omega\) is a nonneagtive function, or a measure \(\omega\in M_+(\Omega)\), here \(M_+(\Omega)\) denotes the class of locally finite positive Borel measures on \(\Omega\). The main goal of this paper is to establish necessary and sufficient conditions for the existence of global solutions of (1.1) on \(\mathbb R^n\;(n\geq 3),\) together with sharp pointwise estimates of solutions and their gradients, without any a priori assumptions on \(\omega\geq 0\). The authors show that all weak solutions of (1.1) belong to a function space intrinsically associated with the equation. And the characterizations of solvability are given explicitly in terms of the pointwise behavior of the corresponding Riesz potentials as well as in geometric terms. The authors prove that analogous results are true for more general semilinear equations of the type \(-Lu=f(x, u, \nabla u)\). In the case of \(q>2\) the solvability of the corresponding Dirichlet problem on a bounded domain \(\Omega\) is also characterized.

35J60 Nonlinear elliptic equations
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
35J10 Schrödinger operator, Schrödinger equation
42B35 Function spaces arising in harmonic analysis
Full Text: DOI
[1] [AH]Adams, D. R. andHedberg, L. I.,Function Spaces and Potential Theory, Springer-Verlag, Berlin-Heidelberg, 1996.
[2] [AP]Adams, D. R. andPierre, M., Capacitary strong type estimates in semilinear problems,Ann. Inst. Fourier (Grenoble) 41:1 (1991), 117–135.
[3] [Ag]Agmon, S., On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, inMethods of Functional Analysis and Theory of Elliptic Equations (Greco, D., ed.), pp. 19–52, Liguori, Naples, 1983.
[4] [A1]Ancona, A., On strong barriers and an inequality of Hardy for domains inR n,J. London Math. Soc. 34 (1986), 274–290. · Zbl 0629.31002 · doi:10.1112/jlms/s2-34.2.274
[5] [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. · Zbl 0944.58016 · doi:10.1007/BF02843153
[6] [B]Baras, P., Semilinear problem with convex nonlinearity, inRecent Advances in Nonlinear Elliptic and Parabolic Problems (Bénilan, P., Chipot, M., Evans, L. C. and Pierre, M., eds.), Pitman Research Notes in Math. Sciences208, pp. 202–215, Longman, Harlow, 1989. · Zbl 0768.35030
[7] [BP]Baras, P. andPierre, M., Singularités éliminables pour des équations semilinéaires,Ann. Inst. Fourier (Grenoble) 34:1 (1984), 185–206.
[8] [CZ]Chung, K. L. andZhao, Z.,From Brownian Motion to Schrödinger’s Equation, Springer-Verlag, Berlin, 1995.
[9] [GW]Grüter, M. andWidman, K.-O., The Green function for uniformly elliptic equations,Manuscripta Math. 37 (1982), 303–342. · Zbl 0485.35031 · doi:10.1007/BF01166225
[10] [H]Hansson, K., Imbedding theorems of Sobolev type in potential theory,Math. Scand.45 (1979), 77–102. · Zbl 0437.31009
[11] [Ha]Hartman, P.,Ordinary Differential Equations, Republ. 2nd ed., Birkhäuser, Boston, Mass., 1982.
[12] [HK]Hayman, W. K. andKennedy, P. B.,Subharmonic Functions, Vol. I, Academic Press, London-New York-San Francisco, 1976.
[13] [He]Hedberg, L. I., On certain convolution inequalities,Proc. Amer. Math. Soc. 36 (1972), 505–510. · Zbl 0283.26003 · doi:10.1090/S0002-9939-1972-0312232-4
[14] [HS]Hueber, H. andSieveking, M., Uniform bounds for quotients of Green functions onC 1,1-domains,Ann. Inst. Fourier (Grenoble) 32:1 (1982), 105–117.
[15] [KV]Kalton, N. J. andVerbitsky, I. E., Nonlinear equations and weighted norm inequalities, to appear inTrans. Amer, Math. Soc.
[16] [L]Lions, P. L., On the existence of positive solutions of semilinear elliptic equations,SIAM Rev. 24 (1982), 441–467. · Zbl 0511.35033 · doi:10.1137/1024101
[17] [M1]Maz’ya, V. G., On the theory of then-dimensional Schrödinger operator,Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1145–1172 (Russian).
[18] [M2]Maz’ya, V. G.,Sobolev Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1985.
[19] [MV]Maz’ya, V. G. andVerbitsky, I. E., Capacitary estimates for fractional integrals, with applications to partial differential equations and Sobolev multipliers,Ark. Mat. 33 (1995), 81–115. · Zbl 0834.31006 · doi:10.1007/BF02559606
[20] [N]Nyström, K., Integrability of Green potentials in fractal domains,Ark. Mat. 34 (1996), 335–381. · Zbl 0860.31002 · doi:10.1007/BF02559551
[21] [S]Schechter, M., Hamiltonians for singular potentials,Indiana Univ. Math. J. 22 (1972), 483–503. · Zbl 0263.47009 · doi:10.1512/iumj.1972.22.22042
[22] [Si]Simon, B., Schrödinger semigroups,Bull. Amer. Math. Soc. 7 (1982), 447–526. · Zbl 0524.35002 · doi:10.1090/S0273-0979-1982-15041-8
[23] [St]Stein, E. M.,Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N. J., 1970. · Zbl 0207.13501
[24] [VW]Verbitsky, I. E. andWheeden, R. L., Weighted inequalities for fractional integrals and applications to semilinear equations,J. Funct. Anal. 129 (1995), 221–241. · Zbl 0830.46029 · doi:10.1006/jfan.1995.1049
[25] [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. · Zbl 0164.13101
[26] [Z]Zhao, Z., Green function for Schrödinger operator and conditioned Feynman-Kac gauge,J. Math. Anal. Appl. 116 (1986), 309–334. · Zbl 0608.35012 · doi:10.1016/S0022-247X(86)80001-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.