The second Yamabe invariant with singularities

Benalili, Mohammed; Boughazi, Hichem

doi:10.5802/ambp.308

Benalili, Mohammed ¹ ; Boughazi, Hichem ¹

¹ Université Aboubekr Belkaïd Faculty of Sciences Dept. of Math. B.P. 119 Tlemcen, Algeria

Annales mathématiques Blaise Pascal, Tome 19 (2012) no. 1, pp. 147-176.

Résumé

Let $(M, g)$ be a compact Riemannian manifold of dimension $n \geq 3$ .We suppose that $g$ is a metric in the Sobolev space $H_{2}^{p} (M, T^{*} M \otimes T^{*} M)$ with $p > \frac{n}{2}$ and there exist a point $P \in M$ 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.

MR Zbl

DOI : 10.5802/ambp.308

Classification : 58J05
Mots clés : Second Yamabe invariant, singularities, Critical Sobolev growth.

Affiliations des auteurs :

Benalili, Mohammed ¹ ; Boughazi, Hichem ¹

¹ Université Aboubekr Belkaïd Faculty of Sciences Dept. of Math. B.P. 119 Tlemcen, Algeria

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Benalili, Mohammed; Boughazi, Hichem. The second Yamabe invariant with singularities. Annales mathématiques Blaise Pascal, Tome 19 (2012) no. 1, pp. 147-176. doi : 10.5802/ambp.308. http://www.numdam.org/articles/10.5802/ambp.308/

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