On the number of critical points of solutions of semilinear elliptic equations

Massimo Grossi; Massimo Grossi

doi:10.3934/era.2021080

Electronic Research Archive

2021, Volume 29, Issue 6: 4215-4228. doi: 10.3934/era.2021080

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On the number of critical points of solutions of semilinear elliptic equations

Massimo Grossi

Dipartimento di Matematica, Università di Roma "La Sapienza", P.le A. Moro 2 - 00185 Roma, Italy

Received: 01 July 2021 Revised: 01 August 2021 Published: 08 October 2021
35B05, 35B06, 35B09

In this survey we discuss old and new results on the number of critical points of solutions of the problem
$ \begin{equation} \begin{cases} -\Delta u = f(u)&in\ \Omega\\ u = 0&on\ \partial \Omega \end{cases} \;\;\;\;\;\;\;\;(0.1)\end{equation} $
where $ \Omega\subset \mathbb{R}^N $ with $ N\ge2 $ is a smooth bounded domain. Both cases where $ u $ is a positive or nodal solution will be considered.
- Critical points,
- degree theory
Citation: Massimo Grossi. On the number of critical points of solutions of semilinear elliptic equations[J]. Electronic Research Archive, 2021, 29(6): 4215-4228. doi: 10.3934/era.2021080

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Abstract

In this survey we discuss old and new results on the number of critical points of solutions of the problem

$ \begin{equation} \begin{cases} -\Delta u = f(u)&in\ \Omega\\ u = 0&on\ \partial \Omega \end{cases} \;\;\;\;\;\;\;\;(0.1)\end{equation} $

where $ \Omega\subset \mathbb{R}^N $ with $ N\ge2 $ is a smooth bounded domain. Both cases where $ u $ is a positive or nodal solution will be considered.

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