  We study the $\Gamma$-convergence of the following functional ($p>2$) $$
F_{\epsilon}(u):=\epsilon^{p-2}\int_{\Omega}|Du|^p d(x,\partial
\Omega)^{a}dx+\frac{1}{\epsilon^{\frac{p-2}{p-1}}}\int_{\Omega}W(u)
d(x,\partial
\Omega)^{-\frac{a}{p-1}}dx+\frac{1}{\sqrt{\epsilon}}\int_{\partial\Omega}V(Tu)d\mathcal{H}^2,
$$ where $\Omega$ is an open bounded set of $\mathbb{R}^3$ and $W$ and $V$ are
two non-negative continuous functions vanishing at $\alpha, \beta$ and
$\alpha', \beta'$, respectively. In the previous functional, we fix $a=2-p$ and
$u$ is a scalar density function, $Tu$ denotes its trace on $\partial\Omega$,
$d(x,\partial \Omega)$ stands for the distance function to the boundary
$\partial\Om$. We show that the singular limit of the energies $F_{\epsilon}$
leads to a coupled problem of bulk and surface phase transitions.
