Γ-Convergence of some super quadratic functionals with singular weights

We study the Γ-convergence of the following functional (p > 2) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{\varepsilon}(u):=\varepsilon^{p-2}\int\limits_{\Omega} |Du|^p d(x,\partial \Omega)^{a}dx+\frac{1}{\varepsilon^{\frac{p-2}{p-1}}} \int\limits_{\Omega} W(u) d(x,\partial \Omega)^{-\frac{a}{p-1}}dx+\frac{1}{\sqrt{\varepsilon}} \int\limits_{\partial\Omega} V(Tu)d\mathcal{H}^2,$$\end{document}where Ω is an open bounded set of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^3}$$\end{document} and W and V are two non-negative continuous functions vanishing at α, β and α′, β′, respectively. In the previous functional, we fix a = 2 − p and u is a scalar density function, Tu denotes its trace on ∂Ω, d(x, ∂Ω) stands for the distance function to the boundary ∂Ω. We show that the singular limit of the energies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F_{\varepsilon}}$$\end{document} leads to a coupled problem of bulk and surface phase transitions.