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Prescribing scalar curvature on \(S^ n\) and related problems. II: Existence and compactness. (English) Zbl 0849.53031
This is a sequel to Part I [J. Differ. Equations 120, No. 2, 319-410 (1995; Zbl 0827.53039)] which studies the prescribing scalar curvature problem on \(S^n\). First we present some existence and compactness results for \(n= 4\). The existence result extends those of A. Bahri and J. M. Coron [J. Funct. Anal. 95, No. 1, 106-172 (1991; Zbl 0722.53032)], M. Benayed, Y. Chen, H. Chtioui and M. Hammami [Duke Math. J. 84, No. 3, 633-677 (1996)] and D. Zhang [New results on geometric variational problems, Thesis, Stanford Univ. (1990)]. The compactness results are new and optimal. In addition, we give a counting formula of all solutions which, as a consequence, gives a complete description of when and where blow-ups occur. It follows from our results that solutions to the problem may have multiple blow-up points. This phenomenon is new and very different from the lower-dimensional cases \(n= 2,3\).
Next we study the problem for \(n\geq 3\). Some existence and compactness results have been given in [the author, loc. cit.] when the order of flatness at critical points of the prescribed scalar curvature functions \(K(x)\) is \(\beta\in (n- 2, n)\). The key point there is that for the class of \(K\) mentioned above we have completed \(L^\infty\) a priori estimates for solutions of the prescribing scalar curvature problem. Here we demonstrate that when the order of flatness at critical points of \(K(x)\) is \(\beta= n- 2\), the \(L^\infty\) estimates for solutions fail in general. In fact, two or more blow-up points occur.
On the other hand, we provide some existence and compactness results when the order of flatness at critical points of \(K(x)\) is \(\beta\in [n- 2, n)\). With this result, we can easily deduce that \(C^\infty\) scalar curvature functions are dense in the \(C^{1, \alpha}\) \((0< \alpha< 1)\) norm among all positive functions. With respect to the \(C^2\) norm, such a density result is false in general.
We also give a simpler proof to a Sobolev-Aubin type inequality established in [S.-Y. A. Chang and P. C. Yang, Duke Math. J. 64, No. 1, 27-69 (1991; Zbl 0739.53027)].
Some of the results in this paper as well as that of Part I have been announced in [C. R. Acad. Sci. Paris, Sér. I 317, No. 2, 159-164 (1993; Zbl 0787.53029)].

MSC:
53C20 Global Riemannian geometry, including pinching
35B40 Asymptotic behavior of solutions to PDEs
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