We consider the semi-Riemannian Yamabe type equations of the form $-\square u+\lambda u=\mu |u{|}^{p-1}u\text{on}M$where $M$ is either the semi-Euclidean space or the pseudosphere of dimension $m\ge 3$, $\square$ is the semi-Riemannian Laplacian in $M$, $\lambda \ge 0$, $\mu \in \mathbb{R}\setminus ︀\left\{0\right\}$ and $p>1$. Using semi-Riemannian isoparametric functions on $M$, we reduce the PDE into a generalized Emden–Fowler ODE of the form $w^{\prime \prime }+q\left(r\right)w^{\prime }+\lambda w=\mu |w{|}^{p-1}w\text{on}I,$where $I\subset \mathbb{R}$ is $[0,\infty )$ or $[0,\pi ]$, $q\left(r\right)$ blows-up at 0 and $w$ is subject to the natural initial conditions $w^{\prime }\left(0\right)=0$ in the first case and $w^{\prime }\left(0\right)=w^{\prime }\left(\pi \right)=0$ in the second. We prove the existence of blowing-up and globally defined solutions to this problem, both positive and sign-changing, inducing solutions to the semi-Riemannian Yamabe type problem with the same qualitative properties, with level and critical sets described in terms of semi-Riemannian isoparametric hypersurfaces and focal varieties. In particular, we prove the existence of sign-changing blowing-up solutions to the semi-Riemannian Yamabe problem in the pseudosphere having a prescribed number of nodal domains.

我们考虑以下形式的半黎曼Yamabe型方程 $-\square ü+\lambda ü=\mu |ü{|}^{p-\mathrm{1个}}ü\text{上}\mathrm{中号}$在哪里 $\mathrm{中号}$ 是半欧空间或维数的伪球 $米\ge 3$， $\square$ 是半黎曼拉普拉斯算子 $\mathrm{中号}$， $\lambda \ge 0$， $\mu \in \mathbb{[R}\setminus ︀\left\{0\right\}$ 和 $p>\mathrm{1个}$。在上使用半黎曼等参函数$\mathrm{中号}$，我们将PDE简化为以下形式的广义Emden-Fowler ODE $w^{\prime \prime }+q（\mathrm{[R}）w^{\prime }+\lambda w=\mu |w{|}^{p-\mathrm{1个}}w\text{上}\mathrm{一世}，$在哪里 $\mathrm{一世}\subset \mathbb{[R}$ 是 $[0，\infty ）$ 或者 $[0，\pi ]$， $q（\mathrm{[R}）$ 炸毁在0和 $w$ 受自然初始条件的约束 $w^{\prime }（0）=0$ 在第一种情况下， $w^{\prime }（0）=w^{\prime }（\pi ）=0$在第二。我们证明了存在正定和正负号变换的此问题的爆炸式和全局定义式解的存在，从而诱导了具有相同定性的半黎曼Yamabe型问题的解，并且以半集形式描述了能级和临界集黎曼等参超曲面和焦点变种。尤其是，我们证明了在具有规定数量的节点域的伪球中，存在对半黎曼Yamabe问题的变号爆炸解决方案的存在。