  The aim of this paper is to investigate two classical problems related to
nilpotent center conditions and bifurcation of limit cycles in switching
polynomial systems. Due to the difficulty in calculating the Lyapunov constants
of switching polynomial systems at non-elementary singular points, it is
extremely difficult to use the existing Poincar\'e-Lyapunov method to study
these two problems. In this paper, we develop a higher-order
Poincar\'e-Lyapunov method to consider the nilpotent center problem in
switching polynomial systems, with particular attention focused on cubic
switching Li\'enard systems. With proper perturbations, explicit center
conditions are derived for switching Li\'enard systems at a nilpotent center,
which is characterized as global. Moreover, with Bogdanov-Takens bifurcation
theory, the existence of five limit cycles around the nilpotent center is
proved for a class of switching Li\'enard systems, which is a new lower bound
of cyclicity for such polynomial systems around a nilpotent center.
