The LOw Frequency ARay (LOFAR) observatory is a multipurpose radio antenna array aimed to detect radio signals in the frequency range 10 - 240 MHz. Radio antennas are clustered into over 50 stations, and are spread along Central and Northern Europe. The LOFAR core, where the density of stations is highest, is instrumented with the LOfar Radboud air shower Array (LORA), covering an area of about 300 m diameter centered at the LOFAR core position. Since 2011 the LOFAR core has been used for detecting radio-signals associated to cosmic-ray air showers in the energy range 1016−10181016−1018 eV. Data acquisition is triggered by the LORA scintillator array, which provides energy, arrival direction, and core position estimates of the detected air shower too. Thus only the core of the LOFAR array is currently used for cosmic-ray detection. In order to extend the energy range of the detected cosmic rays, it is necessary to expand the effective collecting area to the whole LOFAR array. On this purpose, a detailed study about the LOFAR potentialities of working in self-trigger mode, i.e. with the cosmic-ray data acquisition trigger provided by the radio-antenna only, is presented here. A new method based on the intensity and the frequency spectrum for determining the air shower position to be implemented on LOFAR remote stations is presented too.