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

TL;DR

This paper proves the existence of heteroclinic solutions for a class of elliptic equations with double well potentials in an infinite cylindrical domain, considering potentials like Ginzburg-Landau and specific asymptotic conditions on the coefficient function.

Contribution

It establishes the existence of heteroclinic solutions under broad conditions on the potential and the coefficient function, including asymptotic periodicity and boundedness assumptions.

Findings

Existence of heteroclinic solutions for the specified elliptic equations.

Applicable to potentials including Ginzburg-Landau.

Results under both asymptotic periodicity and boundedness conditions.

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 $$ 0<A_0=A(0,y)=\inf_{(x,y) \in \Omega}A(x,y) < \liminf_{|(x,y)| \to +\infty}A(x,y)=A_\infty<\infty, \quad \forall y \in \D. $$