  We analyze the influence of the density dependence of the symmetry energy on
the average excitation energy of the isoscalar giant monopole resonance (GMR)
in stable and exotic neutron-rich nuclei by applying the relativistic extended
Thomas-Fermi method in scaling and constrained calculations. For the effective
nuclear interaction, we employ the relativistic mean field model supplemented
by an isoscalar-isovector meson coupling that allows one to modify the density
dependence of the symmetry energy without compromising the success of the model
for binding energies and charge radii. The semiclassical estimates of the
average energy of the GMR are known to be in good agreement with the results
obtained in full RPA calculations. The present analysis is performed along the
Pb and Zr isotopic chains. In the scaling calculations, the excitation energy
is larger when the symmetry energy is softer. The same happens in the
constrained calculations for nuclei with small and moderate neutron excess.
However, for nuclei of large isospin the constrained excitation energy becomes
smaller in models having a soft symmetry energy. This effect is mainly due to
the presence of loosely-bound outer neutrons in these isotopes. A sharp
increase of the estimated width of the resonance is found in largely
neutron-rich isotopes, even for heavy nuclei, which is enhanced when the
symmetry energy of the model is soft. The results indicate that at large
neutron numbers the structure of the low-energy region of the GMR strength
distribution changes considerably with the density dependence of the nuclear
symmetry energy, which may be worthy of further characterization in RPA
calculations of the response function.
