Existence of a Heteroclinic Solution for a~Double Well Potential Equation in an Infinite Cylinder of ℝ ^N

Abstract

This paper concerns with the existence of a heteroclinic solution for the following class of elliptic equationswhere ǫ > 0, Ω = R × D is an infinite cylinder of R N with N ≥ 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)∈Ω A(x, y) < lim inf |(x,y)|→+∞ A(x, y) = A ∞ < ∞, ∀y ∈ D.