In this paper, basic observational properties of elliptical galaxies such as the integrated spectra, the chemical abundances and the enhancement of alpha-elements inferred from broad-band colors and line strength indices Mg(2) and (Fe) land their gradients), the color-magnitude relation, the UV fluxes, and the mass-luminosity ratios, are examined in the light of current theoretical interpretations, and attention is called on several points of internal contradiction. Indeed existing models for the formation and evolution of elliptical galaxies are not able to simultaneously account for all of the above observational features. Specifically, in the context of standard star formation in the galactic-wind driven models, that are at the base of present-day understanding of the color-magnitude relation and UV fluxes, it is difficult to explain the slope of the M/L(B) versus M(B) relation (tilt of the Fundamental Plane) and enhancement of the alpha-elements in the brightest elliptical galaxies. We suggest that the new initial mass function (IMF) by Padoan et al. (1997), which depends on the temperature, density, and velocity dispersion of the medium in which stars are formed, may alleviate some of the difficulties in question. Models of elliptical galaxies incorporating the new IMF (varying with time and position inside a galaxy) are presented and discussed at some extent. In brief, in a hot, rarefied medium the new IMF is more skewed toward the high mass end than in a cool, dense medium, a situation which is met passing from high to low mass galaxies or from the external regions to the center of a galaxy. As a result of the changing IMF, the enhancement of alpha-elements and tilt of the Fundamental Plane are easily explained leaving unaltered the interpretation of the remaining properties under examination. Finally, some implications concerning the relative proportions of visible stars, collapsed remnants (baryonic dark matter), and gas left over by the star forming process are examined.
