# The second Yamabe invariant with singularities

• [1] Université Aboubekr Belkaïd Faculty of Sciences Dept. of Math. B.P. 119 Tlemcen, Algeria
• Volume: 19, Issue: 1, page 147-176
• ISSN: 1259-1734

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## Abstract

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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.

## How to cite

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Benalili, Mohammed, and Boughazi, Hichem. "The second Yamabe invariant with singularities." Annales mathématiques Blaise Pascal 19.1 (2012): 147-176. <http://eudml.org/doc/251055>.

@article{Benalili2012,
abstract = {Let $(M,g)$ be a compact Riemannian manifold of dimension $n\ge 3$.We suppose that $g$ is a metric in the Sobolev space $H_\{2\}^\{p\}(M,T^\{\ast \}M\otimes T^\{\ast \}M)$ with $p&gt;\frac\{n\}\{2\}$ and there exist a point $\ P\in M$ and $\delta &gt;0$ such that $g$ is smooth in the ball $\ B_\{p\}(\delta )$. 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.},
affiliation = {Université Aboubekr Belkaïd Faculty of Sciences Dept. of Math. B.P. 119 Tlemcen, Algeria; Université Aboubekr Belkaïd Faculty of Sciences Dept. of Math. B.P. 119 Tlemcen, Algeria},
author = {Benalili, Mohammed, Boughazi, Hichem},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Second Yamabe invariant; singularities; Critical Sobolev growth; second Yamabe invariant; critical Sobolev growth},
language = {eng},
month = {1},
number = {1},
pages = {147-176},
publisher = {Annales mathématiques Blaise Pascal},
title = {The second Yamabe invariant with singularities},
url = {http://eudml.org/doc/251055},
volume = {19},
year = {2012},
}

TY - JOUR
AU - Benalili, Mohammed
AU - Boughazi, Hichem
TI - The second Yamabe invariant with singularities
JO - Annales mathématiques Blaise Pascal
DA - 2012/1//
PB - Annales mathématiques Blaise Pascal
VL - 19
IS - 1
SP - 147
EP - 176
AB - Let $(M,g)$ be a compact Riemannian manifold of dimension $n\ge 3$.We suppose that $g$ is a metric in the Sobolev space $H_{2}^{p}(M,T^{\ast }M\otimes T^{\ast }M)$ with $p&gt;\frac{n}{2}$ and there exist a point $\ P\in M$ and $\delta &gt;0$ such that $g$ is smooth in the ball $\ B_{p}(\delta )$. 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.
LA - eng
KW - Second Yamabe invariant; singularities; Critical Sobolev growth; second Yamabe invariant; critical Sobolev growth
UR - http://eudml.org/doc/251055
ER -

## References

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5. Emmanuel Hebey, Introductions à l’analyse sur les variétés, 5 (1997), Diderot Éditeur, Arts et sciences, Paris
6. Farid Madani, Le problème de Yamabe avec singularités, Bull. Sci. Math. 132 (2008), 575-591 Zbl1152.58017MR2455898
7. R. Schoen, S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), 47-71 Zbl0658.53038MR931204
8. Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), 479-495 Zbl0576.53028MR788292
9. Neil S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 265-274 Zbl0159.23801MR240748

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