Saddle solutions for Allen-Cahn type equations involving the prescribed mean curvature operator

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 $t=\pm \alpha$.