This paper discusses the situation that the large deviation rate functional has two distinct minimizers, for a model described by Wiener measures with certain densities involving a scaling. The motivation comes from the study of the so-called $\nabla\varphi$ interface model with weak self potentials. The pinned Wiener measures case was discussed by .
"Law of Large Numbers for Wiener Measure with Density Having Two Large Deviation Minimizers." Tokyo J. Math. 32 (1) 279 - 286, June 2009. https://doi.org/10.3836/tjm/1249648422