The problem of H-infinity control of switched nonlinear systems in p-normal form is investigated in this technical note where the solvability of the H-infinity control problem for individual subsystems is unnecessary. Using the generalized multiple Lyapunov functions method and the adding a power integrator technique, we design a switching law and construct continuous state feedback controllers of subsystems explicitly by a recursive design algorithm to produce global asymptotical stability and a prescribed H-infinity performance level. Multiple Lyapunov functions are exploited to reduce the conservativeness caused by adoption of a common Lyapunov function for all subsystems, which is usually required when applying the backstepping-like recursive design scheme. An example is provided to demonstrate the effectiveness of the proposed design method.