Passing current at given threshold voltages through a metal/insulator/metal sandwich structure device may change its resistive state. Such switching has been rationalized by ion drift models, or changes in electronic states, but the underlying physical mechanism is poorly understood. We propose a new model based on electrostatics to explain multiple resistive states in memristors that contain large defect densities. The different resistive states are due to spontaneously charged states of the insulator 'storage medium', characterized by different 'band bending' solutions of Poisson's equation. For an insulator with mainly donor type defects, the low-resistivity state is characterized by a negatively charged insulator due to convex band bending, and the high-resistivity state by a positively charged insulator due to concave band bending; vice versa for insulators with mainly acceptor type defects. We show that these multiple solutions coexist only for nanoscale devices and for bias voltages limited by the switching threshold values, where the system charge spontaneously changes and the system switches to another resistive state. We outline the general principles how this functionality depends on material properties and defect abundance of the insulator 'storage medium'.