Some counterexamples related to the stationary Kirchhoff equation
Jorge García-Melián, Leonelo Iturriaga · Proceedings of the American Mathematical Society · 2015
In this note we consider the stationary Kirchhoff equation \[ { − M ( ‖ u ‖ 2 ) Δ u = f ( x , u ) a m p ; in Ω , u = 0 a m p ; on ∂ Ω , \left \{ \begin {array}{ll} -M(\| u\|^2) \Delta u = f(x,u) & \hbox {in }\Omega ,\\ \ \ u=0 & \hbox {on }\partial \Omega , \end {array} \right . \] where M M is a continuous positive function and ‖ ⋅ ‖ \| \cdot \| is the standard norm in H 0 1 ( Ω ) 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.