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 \(\begin{aligned} -div\left( \frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\right) + A(\epsilon x,y)V'(u)=0~~\text { in }~~\mathbb {R}^2, \end{aligned}\) $$\begin{array}{c}\hfill -div\left(\frac{\mathrm{\nabla }u}{\sqrt{1+{|\mathrm{\nabla }u|}^2}}\right)+A(ϵx,y)V^{\prime }(u)=0\text{in}{\mathbb{R}}^2,\end{array}$$ where \(\epsilon >0\) $ϵ>0$ and V is a double-well potential with minima at \(t=\alpha \) $t=\alpha$ and \(t=\beta \) $t=\beta$ with \(\alpha <\beta \) $\alpha <\beta$ . 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.