In this paper, we consider the recovery of k-sparse signals using the weighted ℓ_p (0 <; p ≤ 1) minimization when some partial prior information on the support is available. First, we present a unified analysis of restricted isometry constant δ_tk with d <; t ≤ 2d (d ) ≥1 is determined by the prior support information) for sparse signal recovery by the weighted ℓ_p (0 <; p ≤ 1) minimization in both noiseless and noisy settings. This result fills a vacancy on δ_tk with t <; 2, compared with previous works on δ_(a+1)k (a > 1). Second, we provide a sufficient condition on δ_tk with 1 <; t ≤ 2 for the recovery of sparse signals using the ℓ_p (0 <; p ≤ 1) minimization, which extends the existing optimal result on δ_2k in the literature. Last, various numerical examples are presented to demonstrate the better performance of the weighted ℓ_p (0 <; p ≤ 1) minimization is achieved when the accuracy of prior information on the support is at least 50%, compared with that of the ℓ_p (0 <; p ≤1) minimization.