Long-range phase coexistence models with degenerate potentials

This survey offers an overview of recent advances in nonlocal phase transition problems, modeled by Ginzburg--Landau type energies of the form $$\frac{1}{4}\iint_{\R^{2n}\setminus (\R^n \setminus \Omega)^2} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy \;+\; \int_\Omega W(u(x))\,dx.$$ Here,~$W$ is a smooth and possibly \textit{degenerate} double well potential, with a polynomial control on its second derivatives near the wells. The emphasis is on qualitative properties of minimizers and critical points of the energy functional.