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On the structure of the conformal scalar curvature equation on $$\mathbb{R}^ n$$. (English) Zbl 0764.35037
Summary: We conduct a thorough study of the conformal scalar curvature equation $$\Delta u+Ku^ p=0$$ in $$\mathbb{R}^ n$$, $$n\geq 3$$, in case the prescribed scalar curvature function $$K$$ is negative. First, we establish the existence and uniqueness of the maximal positive solution. Then a complete classification of all possible positive solutions is obtained if $$K$$ behaves like $$-| x|^{-\ell}$$ near $$\infty$$ for some constant $$\ell>2$$.

##### MSC:
 35J60 Nonlinear elliptic equations 53A30 Conformal differential geometry (MSC2010) 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) 35J15 Second-order elliptic equations 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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