Becker Social Pricing and Lack of Smooth Demand

I was reading an article on Becker Social Pricing, where your demand for a good depends on other people’s demand for the good. So basically this paper is meant to be a primer into explaining why some places price in such a way that they have chronic shortages (think of big concerts or popular restaurants). I think the idea is that seeing many people want to attend this event actually is part of what contributes to this high demand for the good, and the demand is not convex.

Since demand is not convex, as in Walrasian smooth demand, (if of course I am interpreting this model correctly), then how do utility maximization problems work here, say, if we had two goods that followed this model? Assuming we still have Kuhn-Tucker sufficiency, would we just set up a Lagrangian as normal, or are there some stronger conditions we need to make sure a maximum exists?