This paper concerns the critical points and the level sets of solutions of the Grushin equation in the plane. After exactly establishing descriptions about the critical points of the homogeneous Gruhin-harmonic polynomials and investigating the local geometric properties of the level sets near these critical points, we prove that the critical points of solutions of the Grushin equation are isolated and each critical point has finite multiplicity. We further estimate the numbers of interior critical points of solutions of the Dirichlet boundary value problem.

中文翻译：

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平面上Grushin调和函数的临界点和能级集

本文涉及平面上Grushin方程的临界点和解的水平集。在准确建立了齐次Gruhin调和多项式的临界点的描述并研究了这些临界点附近的能级集的局部几何性质之后，我们证明了Grushin方程解的临界点是孤立的，并且每个临界点都具有多样性。我们进一步估计Dirichlet边值问题解的内部临界点的数量。