Blow-up examples for second order elliptic PDEs of critical Sobolev growth

Druet, Olivier; Hebey, Emmanuel

doi:10.1090/S0002-9947-04-03681-5

Blow-up examples for second order elliptic PDEs of critical Sobolev growth
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by Olivier Druet and Emmanuel Hebey PDF

Trans. Amer. Math. Soc. 357 (2005), 1915-1929 Request permission

Abstract:

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \ge 3$, and $\Delta _g = -div_g\nabla$ be the Laplace-Beltrami operator. Let also $2^\star$ be the critical Sobolev exponent for the embedding of the Sobolev space $H_1^2(M)$ into Lebesgue’s spaces, and $h$ be a smooth function on $M$. Elliptic equations of critical Sobolev growth such as \begin{equation*} (E)\qquad \qquad \qquad \qquad \qquad \qquad \Delta _gu + hu = u^{2^\star -1} \qquad \qquad \qquad \qquad \qquad \qquad \end{equation*} have been the target of investigation for decades. A very nice $H_1^2$-theory for the asymptotic behaviour of solutions of such equations has been available since the 1980’s. The $C^0$-theory was recently developed by Druet-Hebey-Robert. Such a theory provides sharp pointwise estimates for the asymptotic behaviour of solutions of $(E)$. It was used as a key point by Druet to prove compactness results for equations such as $(E)$. An important issue in the field of blow-up analysis, in particular with respect to previous work by Druet and Druet-Hebey-Robert, is to get explicit nontrivial examples of blowing-up sequences of solutions of $(E)$. We present such examples in this article.

References

Haïm Brezis and Jean-Michel Coron, Convergence de solutions de $H$-systèmes et application aux surfaces à courbure moyenne constante, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 16, 389–392 (French, with English summary). MR 748929
H. Brezis and J.-M. Coron, Convergence of solutions of $H$-systems or how to blow bubbles, Arch. Rational Mech. Anal. 89 (1985), no. 1, 21–56. MR 784102, DOI 10.1007/BF00281744
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. MR 982351, DOI 10.1002/cpa.3160420304
Olivier Druet, From one bubble to several bubbles: the low-dimensional case, J. Differential Geom. 63 (2003), no. 3, 399–473. MR 2015469
Olivier Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Not. 23 (2004), 1143–1191. MR 2041549, DOI 10.1155/S1073792804133278
Olivier Druet, Emmanuel Hebey, and Frédéric Robert, A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth, Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 19–25. MR 1988868, DOI 10.1090/S1079-6762-03-00108-2
—, Blow-up theory for elliptic PDEs in Riemannian geometry, Mathematical Notes, Princeton University Press, to appear.
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. MR 1688256
—, Variational methods and elliptic equations in Riemannian geometry, Preprint, Notes from lectures given at ICTP, 2003. Available at http://www.u-cergy.fr/rech/pages/hebey.
Li, Y.Y., and Zhang, L., A Harnack type inequality for the Yamabe equation in low dimensions, Calc. Var. Partial Differential Equations, to appear.
Yanyan Li and Meijun Zhu, Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math. 1 (1999), no. 1, 1–50. MR 1681811, DOI 10.1142/S021919979900002X
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201. MR 834360, DOI 10.4171/RMI/6
Morio Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry 6 (1971/72), 247–258. MR 303464
J. Sacks and K. Uhlenbeck, The existence of minimal immersions of $2$-spheres, Ann. of Math. (2) 113 (1981), no. 1, 1–24. MR 604040, DOI 10.2307/1971131
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. MR 994021, DOI 10.1007/BFb0089180
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. MR 1173050
Michael Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z. 187 (1984), no. 4, 511–517. MR 760051, DOI 10.1007/BF01174186
Henry C. Wente, Large solutions to the volume constrained Plateau problem, Arch. Rational Mech. Anal. 75 (1980/81), no. 1, 59–77. MR 592104, DOI 10.1007/BF00284621