×

Prescribing scalar curvature on \(S^{3}\). (English) Zbl 1151.35360

Summary: We give existence results for solutions of the prescribed scalar curvature equation on \(S^{3}\), when the curvature function is a positive Morse function and satisfies an index-count condition.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Ambrosetti, A.; Badiale, M., Homoclinics: Poincaré-Melnikov type results via a variational approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15, 2, 233-252 (1998) · Zbl 1004.37043
[2] Ambrosetti, A.; Garcia Azorero, J.; Peral, I., Perturbation of \(\Delta u + u^{(N + 2) /(N - 2)} = 0\), the scalar curvature problem in \(R^N\), and related topics, J. Funct. Anal., 165, 1, 117-149 (1999) · Zbl 0938.35056
[3] Aubin, T., Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics (1998), Springer-Verlag: Springer-Verlag Berlin · Zbl 0896.53003
[4] Aubin, T.; Bahri, A., Une hypothèse topologique pour le problème de la courbure scalaire prescrite, J. Math. Pures Appl. (9), 76, 10, 843-850 (1997) · Zbl 0916.58041
[5] Bahri, A.; Coron, J.-M., The scalar-curvature problem on the standard three-dimensional sphere, J. Funct. Anal., 95, 1, 106-172 (1991) · Zbl 0722.53032
[6] Bianchi, G., Non-existence and symmetry of solutions to the scalar curvature equation, Comm. Partial Differential Equations, 21, 1-2, 229-234 (1996) · Zbl 0844.35025
[7] Bourguignon, J.-P.; Ezin, J.-P., Scalar curvature functions in a conformal class of metrics and conformal transformations, Trans. Amer. Math. Soc., 301, 2, 723-736 (1987) · Zbl 0622.53023
[8] Brezis, H.; Kato, T., Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9), 58, 2, 137-151 (1979) · Zbl 0408.35025
[9] Caffarelli, L. A.; Gidas, B.; Spruck, J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42, 3, 271-297 (1989) · Zbl 0702.35085
[10] Chang, S.-Y. A.; Gursky, M. J.; Yang, P. C., The scalar curvature equation on 2- and 3-spheres, Calc. Var. Partial Differential Equations, 1, 2, 205-229 (1993) · Zbl 0822.35043
[11] Chen, C.-C.; Lin, C.-S., Prescribing scalar curvature on \(S^N\). I. A priori estimates, J. Differential Geom., 57, 1, 67-171 (2001) · Zbl 1043.53028
[12] Chen, W.; Li, C., Prescribing scalar curvature on \(S^n\), Pacific J. Math., 199, 1, 61-78 (2001) · Zbl 1060.53047
[13] Gidas, B.; Ni, W. M.; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in \(R^n\), (Mathematical Analysis and Applications, Part A. Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., vol. 7 (1981), Academic Press: Academic Press New York), 369-402
[14] Kazdan, J. L.; Warner, F. W., Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures, Ann. of Math. (2), 101, 317-331 (1975) · Zbl 0297.53020
[15] Li, Y. Y., Prescribing scalar curvature on \(S^n\) and related problems. I, J. Differential Equations, 120, 2, 319-410 (1995) · Zbl 0827.53039
[16] Li, Y. Y., Prescribing scalar curvature on \(S^n\) and related problems. II. Existence and compactness, Comm. Pure Appl. Math., 49, 6, 541-597 (1996) · Zbl 0849.53031
[17] M. Schneider, A priori estimates for the prescribed scalar curvature equation on \(S^3\); M. Schneider, A priori estimates for the prescribed scalar curvature equation on \(S^3\)
[18] Schoen, R.; Zhang, D., Prescribed scalar curvature on the \(n\)-sphere, Calc. Var. Partial Differential Equations, 4, 1, 1-25 (1996) · Zbl 0843.53037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.