The second Yamabe invariant with singularities
Annales Mathématiques Blaise Pascal, Tome 19 (2012) no. 1, pp. 147-176.

Let $\left(M,g\right)$ be a compact Riemannian manifold of dimension $n\ge 3$.We suppose that $g$ is a metric in the Sobolev space ${H}_{2}^{p}\left(M,{T}^{*}M\otimes {T}^{*}M\right)$ with $p>\frac{n}{2}$ and there exist a point $\phantom{\rule{4pt}{0ex}}P\in M$ and $\delta >0$ such that $g$ is smooth in the ball $\phantom{\rule{4pt}{0ex}}{B}_{p}\left(\delta \right)$. We define the second Yamabe invariant with singularities as the infimum of the second eigenvalue of the singular Yamabe operator over a generalized class of conformal metrics to $g$ and of volume $1$. We show that this operator is attained by a generalized metric, we deduce nodal solutions to a Yamabe type equation with singularities.

DOI : https://doi.org/10.5802/ambp.308
Classification : 58J05
Mots clés: Second Yamabe invariant, singularities, Critical Sobolev growth.
@article{AMBP_2012__19_1_147_0,
author = {Mohammed Benalili and Hichem Boughazi},
title = {The second Yamabe invariant with singularities},
journal = {Annales Math\'ematiques Blaise Pascal},
pages = {147--176},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {19},
number = {1},
year = {2012},
doi = {10.5802/ambp.308},
mrnumber = {2978317},
zbl = {1256.58005},
language = {en},
url = {ambp.centre-mersenne.org/item/AMBP_2012__19_1_147_0/}
}
Mohammed Benalili; Hichem Boughazi. The second Yamabe invariant with singularities. Annales Mathématiques Blaise Pascal, Tome 19 (2012) no. 1, pp. 147-176. doi : 10.5802/ambp.308. https://ambp.centre-mersenne.org/item/AMBP_2012__19_1_147_0/

[1] B. Ammann; E. Humbert The second Yamabe invariant, J. Funct. Anal., Volume 235 (2006) no. 2, pp. 377-412 | Article | MR 2225458

[2] Thierry Aubin Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9), Volume 55 (1976) no. 3, pp. 269-296 | MR 431287 | Zbl 0336.53033

[3] Mohammed Benalili; Hichem Boughazi On the second Paneitz-Branson invariant, Houston J. Math., Volume 36 (2010) no. 2, pp. 393-420 | MR 2661253

[4] Haïm Brézis; Elliott Lieb A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., Volume 88 (1983) no. 3, pp. 486-490 | Article | MR 699419 | Zbl 0526.46037

[5] Emmanuel Hebey Introductions à l’analyse sur les variétés, Courant Lecture Notes in Mathematics, Volume 5, Diderot Éditeur, Arts et sciences, Paris, 1997

[6] Farid Madani Le problème de Yamabe avec singularités, Bull. Sci. Math., Volume 132 (2008) no. 7, pp. 575-591 | Article | MR 2455898

[7] R. Schoen; S.-T. Yau Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math., Volume 92 (1988) no. 1, pp. 47-71 | Article | MR 931204 | Zbl 0658.53038

[8] Richard Schoen Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., Volume 20 (1984) no. 2, pp. 479-495 http://projecteuclid.org/getRecord?id=euclid.jdg/1214439291 | MR 788292 | Zbl 0576.53028

[9] Neil S. Trudinger Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3), Volume 22 (1968), pp. 265-274 | Numdam | MR 240748 | Zbl 0159.23801