Let $(M,g)$ be a compact Riemannian manifold of dimension $nge 3$.We suppose that $ g$ is a metric in the Sobolev space $H_{2}^{p}(M,T^{st }Motimes T^{st }M)$ with $ p>rac{n}{2}$ and there exist a point $ Pin M$ and $ delta >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.
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机译：假设$（M，g）$是维度为$ n ge 3 $的紧凑黎曼流形，我们假设$ g $是Sobolev空间$ H_ {2} ^ {p}（M，T ^ { ast} M otimes T ^ { ast} M）$，带有$ p> frac {n} {2} $，并且在M $和$ delta> 0 $中存在一个点$ P ，使得$ g $在球$ B_ {p}（ delta）$中是平滑的。我们将奇异的第二个Yamabe不变量定义为奇异Yamabe运算符的第二个特征值在$ g $的共形量和$ 1 $量的广义类上的最小值。我们证明了该算子是通过广义度量获得的，我们推导了具有奇异性的Yamabe型方程的节点解。
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