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