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A \(C^0\)-theory for the blow-up of second order elliptic equations of critical Sobolev growth. (English) Zbl 1061.58020
Summary: Let \((M,g)\) be a smooth compact Riemannian manifold of dimension \(n \geq 3\), and \(\Delta_g = -div_g\nabla\) the Laplace-Beltrami operator. Also let \(2^\star\) be the critical Sobolev exponent for the embedding of the Sobolev space \(H_1^2(M)\) into Lebesgue spaces, and \(h\) a smooth function on \(M\). Elliptic equations of critical Sobolev growth like \[ \Delta_gu + hu = u^{2^\star-1} \] have been the target of investigation for decades. A very nice \(H_1^2\)-theory for the asymptotic behaviour of solutions of such an equation is available since the 1980’s. In this announcement we present the \(C^0\)-theory we have recently developed. Such a theory provides sharp pointwise estimates for the asymptotic behaviour of solutions of the above equation.

MSC:
58J05 Elliptic equations on manifolds, general theory
53C20 Global Riemannian geometry, including pinching
35J60 Nonlinear elliptic equations
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[1] F. V. Atkinson and L. A. Peletier, Elliptic equations with nearly critical growth, J. Differential Equations 70 (1987), no. 3, 349 – 365. · Zbl 0657.35058 · doi:10.1016/0022-0396(87)90156-2 · doi.org
[2] Haïm Brezis and Lambertus A. Peletier, Asymptotics for elliptic equations involving critical growth, Partial differential equations and the calculus of variations, Vol. I, Progr. Nonlinear Differential Equations Appl., vol. 1, Birkhäuser Boston, Boston, MA, 1989, pp. 149 – 192. · Zbl 0685.35013
[3] Luis A. Caffarelli, Basilis Gidas, and Joel Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271 – 297. · Zbl 0702.35085 · doi:10.1002/cpa.3160420304 · doi.org
[4] Olivier Druet, The best constants problem in Sobolev inequalities, Math. Ann. 314 (1999), no. 2, 327 – 346. · Zbl 0934.53028 · doi:10.1007/s002080050297 · doi.org
[5] Olivier Druet, Sharp local isoperimetric inequalities involving the scalar curvature, Proc. Amer. Math. Soc. 130 (2002), no. 8, 2351 – 2361. · Zbl 1067.53026
[6] -, From one bubble to several bubbles. The low-dimensional case, Preprint, 2002.
[7] Druet, O., and Hebey, E., The \(AB\) program in geometric analysis. Sharp Sobolev inequalities and related problems, Memoirs of the American Mathematical Society, MEMO/160/761, 2002. · Zbl 1023.58009
[8] Druet, O., and Hebey, E., Blow-up examples for second order elliptic PDEs of critical Sobolev growth, Preprint, 2002. · Zbl 1061.58017
[9] Druet, O., Hebey, E., and Robert, F., Blow-up theory for elliptic PDEs in Riemannian geometry, Preprint, 201 pages, 2002. · Zbl 1059.58017
[10] Olivier Druet and Frédéric Robert, Asymptotic profile for the sub-extremals of the sharp Sobolev inequality on the sphere, Comm. Partial Differential Equations 26 (2001), no. 5-6, 743 – 778. · Zbl 0998.58010 · doi:10.1081/PDE-100002377 · doi.org
[11] Zheng-Chao Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), no. 2, 159 – 174 (English, with French summary). · Zbl 0729.35014
[12] Emmanuel Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lecture Notes in Mathematics, vol. 5, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999. · Zbl 0981.58006
[13] -, Nonlinear elliptic equations of critical Sobolev growth from a dynamical viewpoint, Preprint, Conference in honor of H. Brézis and F. Browder, Rutgers university, 2001.
[14] Emmanuel Hebey and Michel Vaugon, The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J. 79 (1995), no. 1, 235 – 279. · Zbl 0839.53030 · doi:10.1215/S0012-7094-95-07906-X · doi.org
[15] Yan Yan Li, Prescribing scalar curvature on \?\(^{n}\) and related problems. I, J. Differential Equations 120 (1995), no. 2, 319 – 410. · Zbl 0827.53039 · doi:10.1006/jdeq.1995.1115 · doi.org
[16] Yanyan Li, Prescribing scalar curvature on \?\(^{n}\) and related problems. II. Existence and compactness, Comm. Pure Appl. Math. 49 (1996), no. 6, 541 – 597. , https://doi.org/10.1002/(SICI)1097-0312(199606)49:63.0.CO;2-A
[17] Frédéric Robert, Asymptotic behaviour of a nonlinear elliptic equation with critical Sobolev exponent: the radial case, Adv. Differential Equations 6 (2001), no. 7, 821 – 846. · Zbl 1087.35026
[18] -, Asymptotic behaviour of a nonlinear elliptic equation with critical Sobolev exponent–The radial case II, Nonlinear Differ. Equ. Appl., 9, 361-384, 2002. · Zbl 1088.35020
[19] Richard M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations (Montecatini Terme, 1987) Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 120 – 154. · doi:10.1007/BFb0089180 · doi.org
[20] Richard M. Schoen, On the number of constant scalar curvature metrics in a conformal class, Differential geometry, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Sci. Tech., Harlow, 1991, pp. 311 – 320. · Zbl 0733.53021
[21] Richard Schoen and Dong Zhang, Prescribed scalar curvature on the \?-sphere, Calc. Var. Partial Differential Equations 4 (1996), no. 1, 1 – 25. · Zbl 0843.53037 · doi:10.1007/BF01322307 · doi.org
[22] Michael Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 (1984), no. 4, 511 – 517. · Zbl 0535.35025 · doi:10.1007/BF01174186 · doi.org
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