Some counterexamples related to the stationary Kirchhoff equation

García-Melián, Jorge; Iturriaga, Leonelo

doi:10.1090/proc/12971

Some counterexamples related to the stationary Kirchhoff equation
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by Jorge García-Melián and Leonelo Iturriaga PDF

Proc. Amer. Math. Soc. 144 (2016), 3405-3411 Request permission

Abstract:

In this note we consider the stationary Kirchhoff equation \[ \left \{ \begin {array}{ll} -M(\| u\|^2) \Delta u = f(x,u) & \hbox {in }\Omega ,\\ \ \ u=0 & \hbox {on }\partial \Omega , \end {array} \right . \] where $M$ is a continuous positive function and $\| \cdot \|$ 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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