Criteria of solvability for multidimensional Riccati equations |
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Authors: | Kurt Hansson Vladimir G. Maz'ya Igor E. Verbitsky |
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Affiliation: | 1. Department of Mathematics, Link?ping University, S-581 83, Link?ping, Sweden 2. Department of Mathematics, University of Missouri, 65211, Columbia, MO, U.S.A.
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Abstract: | We study the solvability problem for the multidimensional Riccati equation ??u=|?u|q+ω, whereq>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 onR 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 , andL is a uniformly elliptic operator. |
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