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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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##### References:
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