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Singularity and decay estimates in superlinear problems via Liouville-type theorems. I: Elliptic equations and systems. (English) Zbl 1146.35038
The goal of the authors is to study some new connections between Liouville-type theorems and local properties of nonnegative solutions to superlinear elliptic problems. By a (nonlinear) Liouville-type theorem the authors mean the statement of nonexistence of nontrivial bounded solutions on the whole space or on a half-space. In the present study the authors develop a general method for derivation of pointwise a priori estimates of local solutions (that is, on an arbitrary domain and without any boundary conditions) from Liouville-type theorems. This leads to new results on universal estimates of spatial singularities for elliptic problems. They also treat semilinear systems of Lane-Emden type. The singularity estimates obtained in this paper seem to be the first result of this type to cover the full subcritical range.
Part II, cf. Indiana Univ. Math. J. 56, No. 2, 879–908 (2007; Zbl 1122.35051).

35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
35J45 Systems of elliptic equations, general (MSC2000)
35J70 Degenerate elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35B33 Critical exponents in context of PDEs
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