This paper considers the exact recovery of $k$-sparse signals in the noiseless setting and support recovery in the noisy case when some prior information on the support of the signals is available. This prior support consists of two parts. One part is a subset of the true support and another part is outside of the true support. For $k$-sparse signals $\mathbf{x}$ with the prior support which is composed of $g$ true indices and $b$ wrong indices, we show that if the restricted isometry constant (RIC) $δ_{k+b+1}$ of the sensing matrix $\mathbf{A}$ satisfies \begin{eqnarray*} δ_{k+b+1}<\frac{1}{\sqrt{k-g+1}}, \end{eqnarray*} then orthogonal matching pursuit (OMP) algorithm can perfectly recover the signals $\mathbf{x}$ from $\mathbf{y}=\mathbf{Ax}$ in $k-g$ iterations. Moreover, we show the above sufficient condition on the RIC is sharp. In the noisy case, we achieve the exact recovery of the remainder support (the part of the true support outside of the prior support) for the $k$-sparse signals $\mathbf{x}$ from $\mathbf{y}=\mathbf{Ax}+\mathbf{v}$ under appropriate conditions. For the remainder support recovery, we also obtain a necessary condition based on the minimum magnitude of partial nonzero elements of the signals $\mathbf{x}$.