Compactness issues and bubbling phenomena for the prescribed Gaussian curvature equation on the Torus

In the spirit of the paper "Large conformal metrics of prescribed Gauss curvature on surfaces of higher genus" by Borer-Galimberti-Struwe, where we dealt with the case of a closed Riemann surface $(M,g_0)$ of genus greater than one, here we study the behaviour of the conformal metrics $g_\lambda$ of prescribed Gauss curvature $K_{g_\lambda} = f_0 + \lambda$ on the torus, when the parameter $\lambda$ tends to one of the boundary points of the interval of existence of $g_\lambda$, and we characterize their "bubbling behaviour".