A method I used for my front wall (behind the false wall) was to use roofing nails, and nail through the fiberglas panels into the wall studs. Roofing nails have big wide heads, so they hold up the panels well. You could do the same for in the soffit, if you hit the joists - and if you can find roofing nails that are long enough. Or, if you have 2x members spanning the bottom of your soffits (like I do - see previous picture), you can put the roofing nails, or nails/screws with fender washers, into those members, with the head or washer hanging off to the side to catch the fiberglas sides. Or you could improvise by using thin metal like 1" x 6", screw the middle into the members, and then the long ends of the metal support the fiberglas.
Maybe post a picture of your soffit, if none of those suggestions seem to help, I'm not sure what yours looks like, how deep, what there is behind it that could support the fiberglas, etc.