The scalar curvature equation on \(\mathbb{R}^ n\) and \(S^ n\)
This paper discusses the equation \(\Delta u+\lambda K(x)u^p=0\) in \(\mathbb{R}^n\) when \(n>2\), \(p\) is the critical exponent \((n+2)/(n-2)\), and \(K\in C^2(\mathbb{R}^n)\) is given. Positive solutions to this equation give a metric on \(\mathbb{R}^n\) which is conformal to the standard one and which has scalar curvature \(K\). Some sufficient conditions are given for this equation to have a solution. The complete description of these conditions is rather complicated, so this review presents a summary. In Theorem 1, near a maximum point of \(K\), the gradient of \(K\) should vanish to a sufficiently high order and, at critical points of \(K\) where \(K\) is sufficiently close to its maximum, \(\Delta K\) is nonzero. In Theorem 2, \(K\) should have only finitely many critical points, there is a strict local maximum point \(z\) of \(K\) at which \(\Delta K\) is nonzero, and, at critical points near \(z\), \(\Delta K\) is nonzero. Unlike most previous works, no symmetry is assumed for \(K\) although recently Noussair and Yang have proved some results for the more general equation \(-\Delta u=K(x)u^p+f(x)\) under different hypotheses on \(K\) which do not require any symmetry. The key to Bianchi's approach is to study the problem on the unit \(n\)-sphere \(S^n\) (as well as the corresponding scalar curvature equation \(\Delta u-{n(n-2)\over 4}u+K(u)u^p=0\)). The proofs involve studying the flow lines of the functional \[ {\mathcal F}(u)={1\over\int_{S^n} K|u|^{p+1}dv}, \] where \(dv\) is the volume element on \(S^n\).
scientific article; zbMATH DE number 896502