  The electronic structure, Fermi surface and elastic properties of the
iso-structural and iso-electronic LaSn$_3$ and YSn$_3$ intermetallic compounds
are studied under pressure within the framework of density functional theory
including spin-orbit coupling. The LaSn$_3$ Fermi surface consists of two
sheets, of which the second is very complex. Under pressure a third sheet
appears around compression $V/V_0=0.94$, while a small topology change in the
second sheet is seen at compression $V/V_0=0.90$. This may be in accordance
with the anomalous behaviour in the superconducting transition temperature
observed in LaSn$_3$, which has been suggested to reflect a Fermi surface
topological transition, along with a non-monotonic pressure dependence of the
density of states at the Fermi level. The same behavior is not observed in
YSn$_3$, the Fermi surface of which already includes three sheets at ambient
conditions, and the topology remains unchanged under pressure. The reason for
the difference in behaviour between LaSn$_3$ and YSn$_3$ is the role of
spin-orbit coupling and the hybridization of La - $4f$ states with the Sn - $p$
states in the vicinity of the Fermi level, which is well explained using the
band structure calculation. The elastic constants and related mechanical
properties are calculated at ambient as well as at elevated pressures. The
elastic constants increase with pressure for both compounds and satisfy the
conditions for mechanical stability under pressure.
