We present a study of molecular hydrogen in a thin surface layer of the disks around five young stars in the Lupus complex. Each system was observed with the Cosmic Origins Spectrograph (COS) onboard the Hubble Space Telescope (HST). We detect a population of fluorescent H2 in all five sources, where the molecules are excited by LyA photons. The location of the emitting gas is constrained to within 15 AU of the central stars because of the temperature required for LyA fluorescence to proceed (T ~ 1500-2500 K). We have used the H2 emission lines to a) estimate the strength of the LyA radiation field that reaches the inner disk and b) extract the radial distribution of hot gas in the inner disk. The objects presented here have very different outer disk morphologies, as seen by ALMA via 890 micron dust continuum emission, ranging from full disks with no signs of cavities to systems with large regions that are clearly depleted (e.g. TYC 7851, with a cavity extending to 75 and 60 AU in dust and gas, respectively). Our results are interpreted in conjunction with sub-mm data from the five systems in an effort to piece together a more complete picture of the overall disk structure.