Heteroclinic connections for fractional Allen-Cahn equations with degenerate potentials

We investigate existence, uniqueness and asymptotic behavior of minimizers of a family of non-local energy functionals of the type $$\frac{1}{4}{\iint }_{{\mathbb{R}}^{2n}\setminus ({\mathbb{R}}^n\setminus \mathrm{\Omega }{)}^2}\mathrm{\mid }u(x)-u(y){\mathrm{\mid }}^{2}K(x-y)dxdy+{\int }_{\mathrm{\Omega }}W(u(x))dx\mathrm{.}$$ 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.