It´s a pretty interesting case the one you describe above. In the context of an Edgeworth-box, I think the restriction you imposed doesn´t inhibite the presence of a contract curve, and the way it is constructed depends on the Ai parameters.
I can think of 3 general cases and some variations to those:
The case where the parameters in Ai are such that the other’s utility is more important in your utility funcions than the logx+ logy part. This means that you are better with the other person benefiting from having more goods that you. Sounds kind of weird, but it is easy to think on an example of this, like the parent-children relationship.
The contract curve will also depend on wether both have this kind of utility function or just one person has it and the other’s utility is defined by a “regular” U function. In the first case, which is the one you describe on your comment, I’m guessing that the contract set would be a curve, just like on a “regular” case and there would be one general equilibrium that depends on the price vector. In the second case, the contract set would be a dot, with the person with the “regular” U function having all of both endowments.
Another case is when the marginal effect of the other’s utility doesn’t compensate the loss of not enjoying the goods. This is an easy one; you might enojoy the idea that every homeless person in the world had a home, but you are not quite enthusiastic on giving away yours. In that case, the contract curve is the one of a “regular” case, same as the equilibrium.
The last case is when the marginal effect of giving the other person more goods is equal to enjoying them, where I’m guessing that the contract set would also be a curv with no singular equilibrium.
You can add some other pecularieties to Ai to make it more interesting, like diminishing returns on utility, but I’ll leave the description of those cases for some other time.
I hope I wasn’t that far from what you were looking for and that you find something useful in my aproach.