  Partially APN functions attract researchers' particular interest recently. It
plays an important role in studying APN functions. In this paper, based on the
multivariate method and resultant elimination, we propose several new infinite
classes of $0$-APN power functions over $\mathbb{F}_{2^n}$. Furthermore, two
infinite classes of $0$-APN power functions $x^d$ over $\mathbb{F}_{2^n}$ are
characterized completely where $(2^k-1)d\equiv 2^m-1~({\rm mod}\ 2^n-1)$ or
$(2^k+1)d\equiv 2^m+1~({\rm mod}\ 2^n-1)$ for some positive integers $n, m, k$.
These infinite classes of $0$-APN power functions can explain some examples of
exponents of Table $1$ in \cite{BKRS2020}.
