As soon as a saddle point is found, people will pay attention to its Morse index. The instability is an important character to a saddle point. For nondegenerate saddle points, the Morse indices can be used to measure their instability and classify them. In this paper, a formula on the Morse index of minimax type saddle point by a Ljusternik–Schnirelman minimax algorithm is established. For nondegenerate minimax type saddle points, the formula clearly answers the question what their Morse indices are. Also, a numerical example on the Hénon equation is presented to illustrate the application of the formula for calculating the Morse indices of numerical solutions in the computation of multiple solutions of the semilinear elliptic equation.

一旦找到鞍点，人们就会关注其莫尔斯指数。不稳定是鞍点的重要特征。对于非退化的鞍点，可以使用莫尔斯指数来测量其不稳定性并将其分类。本文利用Ljusternik–Schnirelman minimax算法建立了minimax型鞍点的摩尔斯指数公式。对于非简并极大极小型鞍点，该公式明确回答了其莫尔斯指数是多少的问题。此外，给出了一个关于Hénon方程的数值示例，以说明该公式在计算半线性椭圆方程的多重解的计算中的应用。