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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 \(\ell>2\).

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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