In this paper, we consider the perturbed Laguerre unitary ensemble described by the weight function of w(x,t)=(x+t) x e with x≥0,t>0,α>0,α+λ+1>0. The Deift–Zhou nonlinear steepest descent approach is used to analyze the limit of the eigenvalue correlation kernel. It was found that under the double scaling s=4nt, n→∞ t→0 such that s is positive and finite, at the hard edge, the limiting kernel can be described by the φ-function related to a third-order nonlinear differential equation, which is equivalent to a particular Painlevé V (shorted as P ) transcendent via a simple transformation. Moreover, this P transcendent is equivalent to a general Painlevé III transcendent. For large s, the P kernel reduces to the Bessel kernel J . For small s, the P kernel reduces to another Bessel kernel J . At the soft edge, the limiting kernel is the Airy kernel as the classical Laguerre weight.