Utility maximization problem : free product vs priced product?
I think there are some problems with the framing of your question. Some points... 

There are some assumptions that need to be added for a meaningful answer. 


Diminishing returns, budget constraint, free disposal etc. 
In terms of utility maximization, there needs to be a budget constraint. if you add one and seek to maximize utility, the consumer will hold $x_2=\frac{w}{p}$ (spend all their money on good 2, and then hold an infinite amount of good 1.


If you are thinking of something like an app, consider non monetary costs. They operate in the same way. Like the time value of playing a worse app. or movies, if you are given a choice to watch a terrible movie for free or pay to see a good movie, you may elect to pay. In which case time is a factor. You can't consume an infinite amount of terrible movies at time $t$.

In summary, you need to add more to the question. If a good is free, marginal utility has to diminish and $\frac{du}{dx}$ needs to equal $0$ at some point or supply has to be finite. Until at least that is satisfied, the consumer will spend all money on on $x_2$ and hold an infinite amount of $x_1$