Existence of heteroclinic solution for a double well potential equation in an infinite cylinder of ${\mathbb{R}}^N$

Abstract

This paper concernes with the existence of heteroclinic solutions for the following class of elliptic equations $$ -Δ{u}+A(εx, y)V'(u)=0, \quad \mbox{in} \quad Ω, $$ where $\epsilon >0$, $Ω=\R \times \D$ is an infinite cylinder of ${\mathbb{R}}^N$ with $N\ge 2$. Here, we have considered a large class of potential $V$ that includes the Ginzburg-Landau potential $V(t)=(t^2-1{)}^2$ and two geometric conditions on the function $A$. In the first condition we assume that $A$ is asymptotic at infinity to a periodic function, while in the second one $A$ satisfies $$ 0<A_0=A(0,y)=\inf_{(x,y) \in Ω}A(x,y) < \liminf_{|(x,y)| \to +\infty}A(x,y)=A_\infty<\infty, \quad \forall y \in \D. $$

In this revised version we have corrected some typos and changed the proof of some lemmas