Abstract In this paper we prove a Morse Lemma for degenerate critical points of a function u which satisfies - Δ ⁢ u = f ⁢ ( u ) in ⁢ B 1 , -\Delta u=f(u)\quad\text{in }B_{1}, where u ∈ C 2 ⁢ ( B 1 ) {u\in C^{2}(B_{1})} , B 1 {B_{1}} is the unit ball of ℝ 2 {\mathbb{R}^{2}} and f is a smooth nonlinearity. Other results on the nondegeneracy of the critical points and the shape of the level sets are proved.

中文翻译：

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ℝ2 中非线性方程解的退化临界点的莫尔斯引理

摘要 在本文中，我们证明了函数 u 的退化临界点的莫尔斯引理，它满足 - Δ ⁢ u = f ⁢ ( u ) in ⁢ B 1 , -\Delta u=f(u)\quad\text{in } B_{1}，其中 u ∈ C 2 ⁢ ( B 1 ) {u\in C^{2}(B_{1})} ，B 1 {B_{1}} 是 ℝ 2 {\mathbb {R}^{2}} 和 f 是平滑非线性。证明了临界点的非退化性和水平集形状的其他结果。