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

Abstract

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