Phase transition problems with the line tension effect: the super-quadratic case

Let $\Omega$ be an open bounded set of $\mathbb{R}^3$ and let $W$ and $V$ be two non-negative continuous functions vanishing at $\alpha, \beta$ and $\alpha', \beta'$, respectively. We analyze the asymptotic behavior as $\varepsilon \to 0$, in terms of $\Gamma$-convergence, of the following functional $$ F_{\varepsilon}(u):=\varepsilon^{p-2}\!\int_{\Omega}\!|Du|^pdx+\frac{1}{\varepsilon^{\frac{p-2}{p-1}}}\!\int_{\Omega}\!W(u)dx+\frac{1}{\varepsilon}\!\int_{\partial\Omega}\!V(Tu)d\mathcal{H}^2 \ \ \ (p>2), $$ where $u$ is a scalar density function and $Tu$ denotes its trace on $\partial\Omega$. We show that the singular limit of the energies $F_{\varepsilon}$ leads to a coupled problem of bulk and surface phase transitions.