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

Let (M,g) be a compact Riemannian manifold of dimension n3.We suppose that g is a metric in the Sobolev space H 2 p (M,T * MT * M) with p>n 2 and there exist a point PM and δ>0 such that g is smooth in the ball B p (δ). 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: 10.5802/ambp.308
Classification: 58J05
Keywords: Second Yamabe invariant, singularities, Critical Sobolev growth.
Mohammed Benalili 1; Hichem Boughazi 1

1 Université Aboubekr Belkaïd Faculty of Sciences Dept. of Math. B.P. 119 Tlemcen, Algeria
@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 = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.308/}
}
TY  - JOUR
AU  - Mohammed Benalili
AU  - Hichem Boughazi
TI  - The second Yamabe invariant with singularities
JO  - Annales mathématiques Blaise Pascal
PY  - 2012
DA  - 2012///
SP  - 147
EP  - 176
VL  - 19
IS  - 1
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.308/
UR  - https://www.ams.org/mathscinet-getitem?mr=2978317
UR  - https://zbmath.org/?q=an%3A1256.58005
UR  - https://doi.org/10.5802/ambp.308
DO  - 10.5802/ambp.308
LA  - en
ID  - AMBP_2012__19_1_147_0
ER  - 
%0 Journal Article
%A Mohammed Benalili
%A Hichem Boughazi
%T The second Yamabe invariant with singularities
%J Annales mathématiques Blaise Pascal
%D 2012
%P 147-176
%V 19
%N 1
%I Annales mathématiques Blaise Pascal
%U https://doi.org/10.5802/ambp.308
%R 10.5802/ambp.308
%G en
%F AMBP_2012__19_1_147_0
Mohammed Benalili; Hichem Boughazi. The second Yamabe invariant with singularities. Annales mathématiques Blaise Pascal, Volume 19 (2012) no. 1, pp. 147-176. doi : 10.5802/ambp.308. https://ambp.centre-mersenne.org/articles/10.5802/ambp.308/

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

[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 | Zbl

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

[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 | DOI | MR | Zbl

[5] Emmanuel Hebey Introductions à l’analyse sur les variétés, Courant Lecture Notes in Mathematics, 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 | DOI | MR

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

[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 | Zbl

[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 | Zbl

Cited by Sources: