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Superlinear elliptic inequalities on manifolds. (English) Zbl 1435.35169

The article are concerned with the existence and nonexistence of solutions for the inequality \(-\Delta u\geq \sigma u^q\) on a connected complete non-compact manifold \(M\). Here \(\sigma\) is a nonnegative Radon measure on \(M\) and \(q>1\) is a real number. The authors deduce necessary and suficient conditions in terms of the Green function that corresponds to the Laplace operator. These conditions are in explicit form when \(M\) has nonnegative Ricci curvature on \(M\) or when the Green function satisfies the \(3G\)-inequality.

MSC:

35J61 Semilinear elliptic equations
58J05 Elliptic equations on manifolds, general theory
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
42B37 Harmonic analysis and PDEs
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