The "Wrong Minimal Surface Equation" does not have the Bernstein property

Abstract

A celebrated result of S. Bernstein states that every solution of the minimal surface equation over the entire plane has to be an affine linear function. Since the paper of Bernstein appeared in 1927, many different proofs and generalizations of this beautiful theorem were given, namely to higher dimensions and to more general equations, for a careful account we refer to the paper by Simo and to the monograph by Dierkes-Hildebrandt-Tromba. In his paper Simon posed the question whether the equation \begin{equation} (1+{u_x}^2)u_{xx}+2u_x u_y u_{xy}+ (1+{u_y}^2)u_{yy} = 0 \label{toll} \end{equation} has the Bernstein property i.e. whether every $C^2$-solution defined over the entire plane necessarily has to be affine. We here show by a very simple argument that this is not the case.