We have investigated the shell structure of nuclei beyond N = 126 relevant to the third peak at A ~ 195 in r-process nuclear abundances. Within the framework of the relativistic Hartree-Bogoliubov theory in deformed space, a comparative study of the response of nuclei with three different Lagrangian models NL-SV1, SVI-2 and NL3 is made. Ground-state properties of even-even nuclei in the isotopic chains of Z = 58 - 72 in the highly neutron-rich region are investigated. A comparison of binding energies with mass formulas shows that in the extreme region of the r-process path, the RMF model SVI-2 without nonlinearities of sigma and omega mesons shows an excellent agreement with the mass formula HFB-24. It is shown that in all the above Lagrangian models, in the neutron rich region between the shell closures N = 126 and N = 184, there is a preponderance of nuclides exhibiting coexistence of well-deformed prolate and oblate shapes in the ground state. A prolate ground state is shown to evolve smoothly to a coexisting oblate ground state via a near degeneracy with successive addition of neutrons. Induced by an onset of deformation for N >= 130, nuclei approaching the r-process path are shown to exhibit enhanced stability in many isotopic chains (Z > 60). This has the effect of shifting the expected neutron drip line significantly farther into the neutron-rich region especially for the higher Z chains. Consequences of the enhanced stability, the shape co-existence and degeneracy of shapes in nuclei on the r-process nucleosynthesis are discussed.