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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.

MSC:
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
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