The goal of this work is to investigate the existence of saddle solutions for the following class of elliptic partial differential equations of the Allen–Cahn type \(\begin{aligned} -div\left( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + A(x,y)V'(u)=0~~\text { in }~~\mathbb {R}^2, \end{aligned}\) where the differential operator is the classical prescribed mean curvature operator. Here, the function \(A:\mathbb {R}^2\rightarrow \mathbb {R}\) exhibits periodicity in all its arguments, while \(V:\mathbb {R}\rightarrow \mathbb {R}\) characterizes a double-well symmetric potential with global minima at \(t=\pm \alpha \) .
