We investigate nodal radial solutions to semilinear problems of type $\left\{\begin{array}{cc}-\Delta u=f(\left|x\right|,u)\hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$where $\Omega$ is a bounded radially symmetric domain of ${\mathbb{R}}^N$ ($N\ge 2$) and $f$ is a real function. We characterize both the Morse index and the degeneracy in terms of a singular one dimensional eigenvalue problem, which is studied in full detail. The presented approach also describes the symmetries of the eigenfunctions. This characterization enables to give a lower bound for the Morse index in a forthcoming work.

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关于奇异特征值问题及其在计算半线性PDE解的摩尔斯指数中的应用

我们研究类型为半线性问题的节点径向解 $\left\{\begin{array}{cc}-\Delta ü=F（\left|X\right|，ü）\hfill & \text{在}\Omega ，\hfill \\ ü=0\hfill & \text{上}\partial \Omega ，\hfill \end{array}\right.$哪里 $\Omega$ 是的有界径向对称域 ${\mathbb{[R}}^{ñ}$ （$ñ\ge 2$）和 $F$是一个真正的功能。我们用一个奇异的一维特征值问题来描述摩尔斯指数和简并性，并对其进行了详细的研究。提出的方法还描述了本征函数的对称性。这种表征使得在即将进行的工作中能够给出莫尔斯指数的下限。