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# A Morse Lemma for Degenerate Critical Points of Solutions of Nonlinear Equations in ℝ2

Massimo Grossi
• Corresponding author
• Dipartimento di Matematica, Università di Roma “La Sapienza”, P.le A. Moro 2, 00185 Roma, Italy
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Published Online: 2019-09-13 | DOI: https://doi.org/10.1515/ans-2019-2055

## Abstract

In this paper we prove a Morse Lemma for degenerate critical points of a function u which satisfies

where $u\in {C}^{2}\left({B}_{1}\right)$, ${B}_{1}$ is the unit ball of ${ℝ}^{2}$ and f is a smooth nonlinearity. Other results on the nondegeneracy of the critical points and the shape of the level sets are proved.

Keywords: Morse Theory; Elliptic Equations; Level Sets

MSC 2010: 35J15

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Accepted: 2019-07-12

Published Online: 2019-09-13

Citation Information: Advanced Nonlinear Studies, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365,

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