  The notion of $c$-differential uniformity has recently received a lot of
attention since its proposal~\cite{Ellingsen}, and recently a characterization
of perfect $c$-nonlinear functions in terms of difference sets in some
quasigroups was obtained in~\cite{AMS22}. Independent of their applications as
a measure for certain statistical biases, the construction of functions,
especially permutations, with low $c$-differential uniformity is an interesting
mathematical problem in this area, and recent work has focused heavily in this
direction. We provide a few classes of permutation polynomials with low
$c$-differential uniformity. The used technique involves handling various Weil
sums, as well as analyzing some equations in finite fields, and we believe
these can be of independent interest.
