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)].
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)].
Reviewer: Y.Li (New Brunswick)
MSC:
| 53C20 | Global Riemannian geometry, including pinching |
| 35B40 | Asymptotic behavior of solutions to PDEs |
Keywords:
prescribing scalar curvature problem; existence result; compactness results; blow-ups; \(L^ \infty\) estimates; Sobolev-Aubin type inequalityReferences:
| [1] | Aubin, Meilleures constantes dans le thérèm d’inclusion de Sobolev st un théorème de Fredholm non linéaire pour la transformation conforme de la courbure scalaire, J. Funct. Anal. 32 pp 149– (1979) · Zbl 0411.46019 · doi:10.1016/0022-1236(79)90052-1 |
| [2] | Aubin, Monge-Ampère Equations (1982) · doi:10.1007/978-1-4612-5734-9 |
| [3] | Bahri, Critical Points at Infinity in Some Variational Problems (1989) · Zbl 0676.58021 |
| [4] | Bahri, The scalar-curvature problem on standard three-dimensional sphere, J. Funct. Anal. 95 pp 106– (1991) · Zbl 0722.53032 · doi:10.1016/0022-1236(91)90026-2 |
| [5] | Bahri, On a variational problem with lack of compactness: The topo-logical effect of the critical points at infinity, Calc. Var. Partial Differential Equations 3 pp 67– (1995) · Zbl 0814.35032 · doi:10.1007/BF01190892 |
| [6] | Also in the preprint series of Centre de Mathématiques 1991 |
| [7] | Benayed, On the prescribed scalar curvature problem on four-manifolds, Duke Math. J. |
| [8] | Besse, Einstein Manifolds (1987) · doi:10.1007/978-3-540-74311-8 |
| [9] | Bianchi, An ODE approach to the equation {\(\Delta\)}u + Ku n+2/n-2 = 0, in Rn, Math. Z. 210 pp 137– (1992) · Zbl 0759.35019 · doi:10.1007/BF02571788 |
| [10] | Bourguignon, Scalar curvature functions in a conformal class of metric and conformal transformations, Tran. Amer. Math. Soc. 301 pp 723– (1987) · doi:10.1090/S0002-9947-1987-0882712-7 |
| [11] | Brezis, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 pp 437– (1983) · Zbl 0541.35029 · doi:10.1002/cpa.3160360405 |
| [12] | Caffarelli, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 pp 271– (1989) · Zbl 0702.35085 · doi:10.1002/cpa.3160420304 |
| [13] | Chang, On Nirenberg’s problem, Internat, J. Math. 4 pp 35– (1993) · Zbl 0786.58010 |
| [14] | Chang, The scalar curvature equation on 2- and 3-sphere, Calc. Var. Partial Differential Equations 1 pp 205– (1993) · Zbl 0822.35043 · doi:10.1007/BF01191617 |
| [15] | Chang, Prescribing Gaussian curvature on S2, Acta Math. 159 pp 215– (1987) · Zbl 0636.53053 · doi:10.1007/BF02392560 |
| [16] | Chang, Conformal deformations of metrics on S2, J. Differential Geom. 27 pp 256– (1988) · Zbl 0649.53022 · doi:10.4310/jdg/1214441783 |
| [17] | Chang, A perturbation result in prescribing scalar curvature on Sn, Duke Math. J. 64 pp 27– (1991) · Zbl 0739.53027 · doi:10.1215/S0012-7094-91-06402-1 |
| [18] | Chen, Scalar curvature on S2, Trans. Amer. Math. Soc. 303 pp 365– (1987) |
| [19] | Chen, A necessary and sufficient condition for the Nirenberg problem, Comm. Pure Appl. Math. 48 pp 657– (1995) · Zbl 0830.35034 · doi:10.1002/cpa.3160480606 |
| [20] | Cheng, Conformal metrics with prescribed Gaussian curvature on S2, Trans. Amer. Math. Soc. 336 pp 219– (1993) |
| [21] | Ding, On the elliptic equation {\(\Delta\)}u + Ku n+2/n-2 = 0 and related topics, Duke Math. J. 52 pp 485– (1985) · Zbl 0592.35048 · doi:10.1215/S0012-7094-85-05224-X |
| [22] | Escobar, Conformal metrics with prescribed scalar curvature, Invent. Math. 86 pp 243– (1986) · Zbl 0628.53041 · doi:10.1007/BF01389071 |
| [23] | Gidas, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 pp 525– (1981) · Zbl 0465.35003 · doi:10.1002/cpa.3160340406 |
| [24] | Gilbarg, Elliptic Partial Differential Equations (1983) |
| [25] | Han, Prescribing Gaussian curvature on S2, Duke Math. J. 61 pp 679– (1990) · Zbl 0715.53034 · doi:10.1215/S0012-7094-90-06125-3 |
| [26] | Han, A note on the Kazdan-Warner type condition, Ann. Inst. H. Poincaré Anal. Non Linéaire |
| [27] | Hebey, Meilleures constantes dans le théorème d’inclusion de Sobolev et multiplicité pour les problèmes de Nirenberg et Yamabe, Indiana Univ. Math. J. 41 pp 377– (1992) · Zbl 0764.53029 · doi:10.1512/iumj.1992.41.41021 |
| [28] | Kazdan, Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature, Ann. Math. 101 pp 317– (1975) · Zbl 0297.53020 · doi:10.2307/1970993 |
| [29] | Li, On -{\(\Delta\)}u = K(x)u5 in R3, Comm. Pure Appl. Math. 46 pp 303– (1993) · Zbl 0799.35068 · doi:10.1002/cpa.3160460302 |
| [30] | Li, Prescribing scalar curvature on Sn and related problems, C. R. Acad. Sci. Paris Sér. I Math. 317 pp 159– (1993) |
| [31] | Li, Prescribing scalar curvature on Sn and related problems, Part I, J. Differential Equations 120 pp 319– (1995) · Zbl 0827.53039 · doi:10.1006/jdeq.1995.1115 |
| [32] | Moser, Dynamical Systems (Proc. Sympos., Univ. Bahia, Salvador) pp 273– (1973) |
| [33] | Nirenberg, Topics in Nonlinear Functional Analysis (1974) |
| [34] | Obata, The conjecture on conformal transformations of Riemannian manifolds, J. Differential Geom. 6 pp 247– (1971) · Zbl 0236.53042 · doi:10.4310/jdg/1214430407 |
| [35] | Onofri, On the positivity of the effective action in a theory of random surface, Comm. Math. Phys. 86 pp 321– (1982) · Zbl 0506.47031 · doi:10.1007/BF01212171 |
| [36] | Pohozaev, Eigenfunctions of the equation {\(\Delta\)}u + {\(\lambda\)}f(u) = 0, Soviet Math. Dokl. 6 pp 1408– (1965) |
| [37] | Schoen, Topics in Calculus of Variations pp 120– (1989) · doi:10.1007/BFb0089180 |
| [38] | Schoen, Pitman Monographs Surveys Pure Appl. Math. 52, in: Differential Geometry pp 311– (1991) |
| [39] | Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 pp 265– (1968) · Zbl 0159.23801 |
| [40] | Zhang , D. New Results on Geometric Variational Problems 1990 |
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