Heteroclinic solutions for some classes of prescribed mean curvature equations in whole $$\mathbb {R}^2$$

The purpose of this paper consists in using variational methods to establish the existence of heteroclinic solutions for some classes of prescribed mean curvature equations of the type

$$\begin{aligned} -div\left( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + A(\epsilon x,y)V'(u)=0~~\text { in }~~\mathbb {R}^2, \end{aligned}$$

where \(\epsilon >0\) and V is a double-well potential with minima at \(t=\alpha \) and \(t=\beta \) with \(\alpha <\beta \). Here, we consider some class of functions A(x, y) that are oscillatory in the variable y and satisfy different geometric conditions such as periodicity in all variables or asymptotically periodic at infinity.