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An ODE approach to the equation \(\Delta u + Ku^{{n+2}\over{n-2}}=0\) in \(\mathbb{R}^ n\). (English) Zbl 0759.35019

The authors consider the problem of finding positive radial solutions of the equation \(\Delta u+Ku^{{n+2}\over {n-2}}=0\) in \(\mathbb{R}^ n\), with \(u(x)=O(| x|^{2-n})\) as \(x\to\infty\). Here \(n>2\) and \(K\) is a radial real valued function. This problem is equivalent to the Kazdan Warner problem on the sphere. Different conditions on \(K\) are given under which there is or is not a solution of the problem. The nonexistence result is different from those obtained from the standard Pokhozhaev identity.

MSC:

35J60 Nonlinear elliptic equations
53A30 Conformal differential geometry (MSC2010)
53C20 Global Riemannian geometry, including pinching
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