Prescribing scalar curvature on \(S^ n\).
Given a function \(R(x)\) on \(S^2\), the well-known Nirenberg problem is to find conditions on \(R(x)\), so that it can be realized as the Gaussian curvature of some conformally related metric. This is equivalent to solving the following nonlinear elliptic equation \[ -\Delta u+1=R(x) e^{2u}\quad x\in S^2. \] On a higher-dimensional sphere \(S^n\), a similar problem was raised by Kazdan and Warner: Which functions \(R(x)\) can be realized as the scalar curvature of some conformally related metrics? This problem is equivalent to consider the existence of a solution to the following nonlinear elliptic equation \[ -\Delta u+\frac{n(n-2)}{4}u= \frac {n-2}{4(n-1)} R(x)u^{\frac{n+2}{n-2}}, \quad u>0,\;x\in S^n.\tag{1} \] The authors prove the following three theorems. Theorem 1. Let \(n\geq 3\). Let \(R=R(r)\) be rotationally symmetric satisfying the following flatness condition near every positive critical point \(r_0\): \(R(r)= R(r_0)+a| r-r_0|^\alpha+ h(| r-r_0|)\), with \(a\neq 0\) and \(n-2<\alpha<n\), where \(h'(s)=o(s^{\alpha-1})\). Then a necessary and sufficient condition for equation (1) to have a solution is that \(R'(r)\) changes signs in the regions where \(R>0\). Theorem 2. There exists a family of rotationally symmetric functions \(R\) for which equation (1) is solvable, however none of the solutions are rotationally symmetric. Theorem 3. Assume that \(R(x)\) has at least two positive local maxima and satisfies the following flatness condition: For any positive critical point \(x_0\) of \(R\), there exists \(\alpha=\alpha(x_0)\in(n-2,n)\), such that in some geodesic normal coordinate system centered at \(x_0\), \[ R(x)= R(0)+\sum^n_{i=1}a_i | x^i|^\alpha+h(x) \] where \(a_i=a_i (x_0)\neq 0\), \(\sum a_i\neq 0\), and \(\nabla h(x)=o(| x|^{\alpha-1})\). Further assume that for any positive critical point \(x_0\) below the two least positive local maxima, holds \(\sum^n_{i=1}a_i(x_0)>0\). Then equation (1) has at least one solution.
scientific article; zbMATH DE number 1867896