In this paper we provide density estimates for a class of functions which includes all the minimizers of the energy \(\begin{aligned} \mathcal {E}_s^p(u,\Omega ):= & (1-s)\left( \frac{1}{2}\int _{\Omega }\int _{\Omega }\frac{\left| u(x)-u(y)\right| ^p}{\left| x-y\right| ^{n+sp}}\,dx\,dy +\int _{\Omega }\int _{\mathbb {R}^n\setminus \Omega }\frac{\left| u(x)-u(y)\right| ^p}{\left| x-y\right| ^{n+sp}}\,dx\,dy\right) \\ & +\int _{\Omega }W(u(x))\,dx, \end{aligned}\) where \(p\in (1,+\infty )\) , \(s \in \left( 0,1\right) \) and W is a double-well potential with polynomial growth \(m\in \left[ p,+\infty \right) \) from the minima. The nonlocal estimates obtained are uniform as \(s\rightarrow 1\) . Moreover, making use of a \(\Gamma \) -convergence result for \(\mathcal {E}_s^p\) as \(s\rightarrow 1\) , we obtain density estimates for the minimizers of the limit energy functional, which takes the form \(\begin{aligned} \mathcal {E}_1^p(u,\Omega ):=\frac{K_{n,p}}{2p}\int _{\Omega } \left| \nabla u(x)\right| ^p+\int _{\Omega } W(u(x))\,dx, \end{aligned}\) for a suitable \(K_{n,p}\in (0,+\infty )\) .
