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Nonexistence and uniqueness of positive solutions of Yamabe type equations on nonpositively curved manifolds. (English) Zbl 0892.53019
Let \((M,g)\) be a complete, noncompact Riemannian manifold of dimension \(m \geq 3\). The authors consider the elliptic equation (resp. inequality) \[ \Delta u+a(x)u-K(x)u^\sigma= 0\quad(\geq 0), \] where \(a(x)\), \(K(x)\) are assigned functions and \(\sigma=\) constant \(>1\). They prove that, if the sectional curvature of \((M,g)\) is nonpositive and under conditions involving the asymptotic behavior of \(a\) and \(K\), the above inequality has no positive \(C^2\)-solution on \(M\). Moreover, they prove, under similar assumptions, the uniqueness of positive \(C^2\)-solutions of the considered elliptic equation.

MSC:
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J05 Elliptic equations on manifolds, general theory
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