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

Abstract

Abstract This paper is concerned with the existence of a heteroclinic solution for the following class of elliptic equations: - Δ ⁢ u + A ⁢ ( ϵ ⁢ x , y ) ⁢ V ′ ⁢ ( u ) = 0   in  ⁢ Ω , -\Delta{u}+A(\epsilon x,y)V^{\prime}(u)=0\quad\mbox{in }\Omega, where ϵ > 0 {\epsilon>0} , Ω = ℝ × 𝒟 {\Omega=\mathbb{R}\times\mathcal{D}} is an infinite cylinder of ℝ N {\mathbb{R}^{N}} with N ≥ 2 {N\geq 2} . Here, we consider a large class of potentials V that includes the Ginzburg–Landau potential V ⁢ ( t ) = ( t 2 - 1 ) 2 {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 ∞ < ∞   for all  ⁢ y ∈ 𝒟 . 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\text{for all }y\in\mathcal{D}.