In this paper, we construct the local minimum of a certain variational problem which we take in the form $\mathrm{inf}\int_\Omega\left\{\frac{\epsilon}{2}kg^2|\nabla w|^2+\frac{1}{4\epsilon}f^2g^4(1-w^2)^2\right\}\,\mathrm{d}x$, where $\epsilon$ is a small positive parameter and $\Omega\subset\mathbb{R}^n$ is a convex bounded domain with smooth boundary. Here $f,g,k\in C^3(\Omega)$ are strictly positive functions in the closure of the domain $\bar{\Omega}$. If we take the inf over all funct...