Permutation polynomials with low c-differential uniformity had wide applications in cryptography and design theory. In this paper, by utilizing the Weil sums technique and solving some certain equations over \(\mathbb {F}_{p^{2m}}\) , we concentrate on characterizing five classes of perfect c-nonlinear (PcN) permutation polynomials of the form \((x^{p^m}-x+\delta )^s+x\) over finite fields with odd characteristic. Firstly, two classes of PcN permutation polynomials are obtained over \(\mathbb {F}_{3^n}\) . Secondly, we characterize the permutation property of a class of polynomials with aforementioned form by using the AGW criterion. Finally, three classes of PcN permutation polynomials are determined over \(\mathbb {F}_{p^{2m}}\) with odd characteristic.
