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Rigidity results for stable solutions of symmetric systems


Author: Mostafa Fazly
Journal: Proc. Amer. Math. Soc. 143 (2015), 5307-5321
MSC (2010): Primary 35J60, 35B35, 35B32, 35D10, 35J20
DOI: https://doi.org/10.1090/proc/12647
Published electronically: June 3, 2015
MathSciNet review: 3411148
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Abstract: We study stable solutions of the nonlinear system

$\displaystyle -\Delta u = H(u)$$\displaystyle \quad \text {in} \ \ \Omega $

where $ u:\mathbb{R}^n\to \mathbb{R}^m$, $ H:\mathbb{R}^m\to \mathbb{R}^m$ and $ \Omega $ is a domain in $ \mathbb{R}^n$. We introduce the novel notion of symmetric systems. The above system is said to be symmetric if the matrix of gradient of all components of $ H$ is symmetric. It seems that this concept is crucial to prove Liouville theorems, when $ \Omega =\mathbb{R}^n$, and regularity results, when $ \Omega =B_1$, for stable solutions of the above system for a general nonlinearity $ H \in C^1(\mathbb{R} ^m)$. Moreover, we provide an improvement for a linear Liouville theorem given by Fazly and Ghoussoub in 2013 that is a key tool to establish De Giorgi type results in lower dimensions for elliptic equations and systems.

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Additional Information

Mostafa Fazly
Affiliation: Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: fazly@ualberta.ca

DOI: https://doi.org/10.1090/proc/12647
Keywords: Elliptic systems, Liouville theorems, stable solutions, radial solutions, regularity theory
Received by editor(s): March 29, 2014
Received by editor(s) in revised form: October 5, 2014
Published electronically: June 3, 2015
Additional Notes: The author is pleased to acknowledge the support of the University of Alberta Start-up Grant RES0019810 and the National Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant RES0020463.
Communicated by: Catherine Sulem
Article copyright: © Copyright 2015 American Mathematical Society