Knapsack combining sum and product
I don't know any reference, so just an extended comment:

If you allow a succinct representation for the input integers, i.e. you can represent the input integers as factors, then the "multiplicative" versions of Knapsack are NP-complete; indeed: 

$\sum_{i=1}^n w_i x_i \leq W$ can be reduced to the equivalent $\prod_{i=1}^n 2^{w_i} x_i \leq 2^W$;

and $\sum_{i=1}^n v_i x_i$ maximization can be reduced to  $\prod_{i=1}^n 2^{v_i} x_i$ maximization.

I didn't think about it too much but the above reduction should work even if we require that in a mixed instance ( maximize sum of values under multiplicative weight constraint OR maximize product of values under addictive weight constraint ) all input values must be given with their prime factorization (building a direct reduction from exact cover).

Without a succinct representation the above reduction doesn't work because Knapsack is not strongly NPC.