Yamabe-type and scalar curvature equations. (English) Zbl 0804.53053
Komatsu, Gen (ed.) et al., Complex geometry. Proceedings of the Osaka international conference, held in Osaka, Japan, Dec. 13-18, 1990. New York: Marcel Dekker. Lect. Notes Pure Appl. Math. 143, 1-12 (1993).
On an \(n\)-dimensional compact Riemannian manifold \((M^ n,g)\) with positive scalar curvature one considers the equations:
\[
(1)\qquad\begin{cases} -Lu= k(x)u^{(n+2)/(n-2)}\\ u\geq 0\text{ on } M^ n\end{cases} \qquad (2)\qquad\begin{cases} -Lu+ qu= k(x)u^{(n+2)/(n-2)}\\ u\geq 0\text{ on } M^ n,\;q\in L(M)\end{cases}
\]
\[ (3)\qquad\begin{cases} -\Delta u= k(x)u^{(n+2)/(n- 2)}\\ u\geq 0\text{ in }\Omega\subseteq \mathbb{R}^ n; \Omega\text{ bounded},\text{ regular }\end{cases} \] called “Yamabe-type equations”. The idea of the article is to try to unify the techniques of solving these equations. For the case \((S^ n,c)\) the common features of these equations as well as a summary of the results are presented. The last part of the article argues, on the base of the homology group the existence of a solution for (1), (2) in the case when \(k(x)= 1\).
For the entire collection see [Zbl 0771.00034].
\[ (3)\qquad\begin{cases} -\Delta u= k(x)u^{(n+2)/(n- 2)}\\ u\geq 0\text{ in }\Omega\subseteq \mathbb{R}^ n; \Omega\text{ bounded},\text{ regular }\end{cases} \] called “Yamabe-type equations”. The idea of the article is to try to unify the techniques of solving these equations. For the case \((S^ n,c)\) the common features of these equations as well as a summary of the results are presented. The last part of the article argues, on the base of the homology group the existence of a solution for (1), (2) in the case when \(k(x)= 1\).
For the entire collection see [Zbl 0771.00034].
Reviewer: S.Noaghi (Vulcan)
MSC:
| 53C20 | Global Riemannian geometry, including pinching |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
