I have an equation that I think should not have too many solutions, but I don't see a way to argue this.

Given $a, b, c, N \in \mathbb{N}$, how many positive integer solutions $x, y \leq N$ can the following equation have.

$$a (x^2 - x) = b(y^2 - y) + c$$

Can we possibly get an asymptotic bound in terms of $N$?