  The goal of this paper is to investigate the existence of saddle solutions
for some classes of elliptic partial differential equations of the Allen-Cahn
type, formulated as follows:
  \begin{equation*}
  -div\left(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) +
A(x,y)V'(u)=0~~\text{ in }~~\mathbb{R}^2.
  \end{equation*}
  Here, the function $A:\mathbb{R}^2\to\mathbb{R}$ exhibits periodicity in all
its arguments, while $V:\mathbb{R}\to\mathbb{R}$ characterizes a double-well
symmetric potential with minima at $t=\pm\alpha$.
