A singular perturbation result with a fractional norm

Let $I$ be an open bounded interval of $\mathbb{R}$ and $W$ a non-negative continuous function vanishing only at $\alpha, \beta \in \mathbb{R}$. We investigate the asymptotic behaviour in terms of $\Gamma$-convergence of the following functional $$ \dys G_{\epsilon}(u):=\epsilon^{p-2}\!\!\int\!\!\!\int_{I\times I}\!\left|\frac{u(x)-u(y)}{x-y}\right|^{p}\!\!dxdy+\frac{1}{\epsilon}\!\!\int_{I}\!W(u)\,dx \ \ (p>2), $$ as $\epsilon\to0$.