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 $$ -\Delta{u}+A(\epsilon x, y)V'(u)=0, \quad \mbox{in} \quad \Omega, $$ where $\epsilon >0$, $\Omega=\R \times \D$ is an infinite cylinder of $\mathbb{R}^N$ with $N \geq 2$. 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 $$ 0<A_0=A(0,y)=\inf_{(x,y) \in \Omega}A(x,y) < \liminf_{|(x,y)| \to +\infty}A(x,y)=A_\infty<\infty, \quad \forall y \in \D. $$