  In this paper, we study the singularly perturbed Laguerre unitary ensemble $$
\frac{1}{Z_n} (\det M)^\alpha e^{- \textrm{tr}\, V_t(M)}dM, \qquad \alpha >0,
$$ with $V_t(x) = x + t/x$, $x\in (0,+\infty)$ and $t>0$. Due to the effect of
$t/x$ for varying $t$, the eigenvalue correlation kernel has a new limit
instead of the usual Bessel kernel at the hard edge 0. This limiting kernel
involves $\psi$-functions associated with a special solution to a new
third-order nonlinear differential equation, which is then shown equivalent to
a particular Painlev\'e III equation. The transition of this limiting kernel to
the Bessel and Airy kernels is also studied when the parameter $t$ changes in a
finite interval $(0, d]$. Our approach is based on Deift-Zhou nonlinear
steepest descent method for Riemann-Hilbert problems.
