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Let (M,g) be a compact manifold of dimension n greater or equals to 3. Wesuppose that g is a given metric in a precised Sobolev space and there is apoint P in M and d>o such that g is smooth on the ball B(P,d). We define thesecond Yamabe invariant with singularities a the minimum of the secondeigenvalue of the singular Yamabe operator over a generalized class ofconformal metrics to g and of volume 1. We show that this operator is attainedby a generalized metric, we deduce nodal solutions to a Yamabe type equationwith singularities.

The second Yamabe invariant with singularities