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Some counterexamples related to the stationary Kirchhoff equation


Authors: Jorge García-Melián and Leonelo Iturriaga
Journal: Proc. Amer. Math. Soc. 144 (2016), 3405-3411
MSC (2010): Primary 35J25, 35J60
DOI: https://doi.org/10.1090/proc/12971
Published electronically: January 26, 2016
MathSciNet review: 3503708
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Abstract: In this note we consider the stationary Kirchhoff equation

$\displaystyle \left \{ \begin {array}{ll} -M(\Vert u\Vert^2) \Delta u = f(x,u) ... ...ox {in }\Omega ,\\ \ \ u=0 & \hbox {on }\partial \Omega , \end{array} \right . $

where $ M$ is a continuous positive function and $ \Vert \cdot \Vert$ is the standard norm in $ H_0^1(\Omega )$. We show that the equation does not enjoy the usual comparison principles (both weak or strong) nor the sub and supersolutions method.

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Jorge García-Melián
Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271, La Laguna, Spain — and — Instituto Universitario de Estudios Avanzados (IUdEA) en Física Atómica, Molecular y Fotónica, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38203, La Laguna, Spain
Email: jjgarmel@ull.es

Leonelo Iturriaga
Affiliation: Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla V-110, Avda. España, 1680, Valparaíso, Chile
Email: leonelo.iturriaga@usm.cl

DOI: https://doi.org/10.1090/proc/12971
Received by editor(s): July 20, 2015
Received by editor(s) in revised form: September 25, 2015
Published electronically: January 26, 2016
Additional Notes: The first author was partially supported by Ministerio de Ciencia e Innovación and Ministerio de Economía y Competitividad under grant MTM2011-27998 (Spain) and Conicyt MEC number 80130002 (Chile)
The second author was supported by Fondecyt number 1120524 and USM Grant 12.15.68 (Chile)
Communicated by: Joachim Krieger
Article copyright: © Copyright 2016 American Mathematical Society

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