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.
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