Density estimates for a nonlocal variational model with a degenerate double-well potential via the Sobolev inequality

We provide density estimates for level sets of minimizers of the energy $\frac{1}{2} \int_{\Omega}\int_{\Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}dxdy+\int_{\Omega}\int_{\mathbb{R}^n\setminus\Omega} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}dxdy+\int_{\Omega}W(u(x))dx$ where $p\in(1,+\infty)$ and $s\in\left(0,\frac{1}{p}\right)$ and $W$ is a double-well potential with polynomial growth $m\in [p,+\infty)$ from the minima. These kinds of potentials are ''degenerate'', since they detach ''slowly'' from the minima, therefore they provide additional difficulties if one wishes to determine the relative sizes of the ''layers'' and the ''pure phases''. To overcome these challenges, we introduce new barriers allowing us to rely on the fractional Sobolev inequality and on a suitable iteration method. The proofs presented here are robust enough to consider the case of quasilinear nonlocal equations driven by the fractional $p$-Laplacian, but our results are new even for the case $p=2$.