We present a model of the electronic transport in a nanotube quantum dot coupled with local phonon modes and with the leads. The Hamiltonian model is analysed by means of a variational approach that is accurate in the whole regime of the coupling parameters. The method, while reproducing the standard solution for limit values of the coupling parameters (e.g. the adiabatic limit), includes the effects of dynamical distortion and the fluctuations of phonon modes. These effects are important to obtain a more accurate approximation of the system quantum state away from these limits. Unusual electrical features, i.e. negative differential resistance, switching and hysteresis, emerge from the current–voltage characteristic of this model molecular device. However, the quantum corrections modify substantially the transport scenario away from the adiabatic limit. In particular, fluctuations and polaronic distortion contrast …