Heteroclinic connections for fractional Allen-Cahn equations with degenerate potentials

Abstract

We investigate existence, uniqueness and asymptotic behavior of minimizers of a family of non-local energy functionals of the type $$\frac{1}{4}{\iint }_{{\mathbb{R}}^{2n}\setminus ({\mathbb{R}}^n\setminus \Omega {)}^2}|u(x)-u(y){|}^{2}K(x-y)dxdy+{\int }_{\Omega }W(u(x))dx.$$ Here, $W$ is a possibly degenerate double well potential with a polynomial control on its second derivative near the wells. Also, $K$ belongs to a wide class of measurable kernels and is modeled on that of the fractional Laplacian.