The second Yamabe invariant with singularities
Annales mathématiques Blaise Pascal, Volume 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: 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
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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

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