Some existence results for the scalar curvature problem via Morse theory
The author states two existence theorems of positive solutions for the equation on the \(n\)-sphere \(S^n\), arising in the prescribed scalar curvature problem. Using the stereographic projection, the problem reduces to find positive solutions \(u\) for the equation \[ -4\;\frac{n-1}{n-2}\Delta u=(1+\varepsilon K(x))u^{\frac{n+2}{n-2}}\text{ on }\mathbb{R}^n, \] where \(\Delta\) is the Laplace-Beltrami operator on \(\mathbb{R}^n\), and \(\varepsilon\) is a small parameter. The proofs rely on an abstract perturbation result developed by \textit{A. Ambrosetti} and \textit{M. Badiale} [Ann. Inst. Henri Poincaré, Analyse Non Linéaire 15, 233-252 (1998; Zbl 1004.37043); announcement in C. R. Acad. Sci., Paris, Sér. I 323, 753-758 (1996; Zbl 0887.34042)], which leads to study a reduced, finite dimensional functional \(\Gamma\). The author shows that Morse theory under general boundary conditions [see \textit{K. C. Chang}, Infinite dimensional Morse theory and multiple solution problems, Birkhäuser, Boston (1993; Zbl 0779.58005)] applies to \(\Gamma\), and allows him to obtain the announced results. When \(n=2\), one of the author's results can be viewed as a partial extension of a result of \textit{S.-Y. A. Chang} and \textit{P. C. Yang} [J. Differ. Geom. 27, 259-296 (1988; Zbl 0649.53022)]; see also \textit{R. Schoen} and \textit{D. Zhang} [Calc. Var. Partial Differ. Equ. 4, 1-25 (1996; Zbl 0843.53037)] for \(n=3\).
scientific article; zbMATH DE number 1822116