  Supersolid, a fascinating quantum state of matter, features novel phenomena
such as the non-classical rotational inertia and transport anomalies. It is a
long standing issue of the coexistence of superfluidity and broken
translational symmetry in condensed matter physics. By recent experimental
advances to create tunable synthetic spin-orbit coupling in ultracold gases,
such highly controllable atomic systems would provide new possibilities to
access supersolidity with no counterpart in solids. Here we report that the
combination of anharmonicity of trapping potential and spin-orbit coupling will
provide a new paradigm to achieve supersolids. By means of imaginary time
evolution of the Gross-Pitaevskii equation, we demonstrate that a supersolid
state can be found when considering a trapped Rashba-type spin-orbit coupled
bosonic atoms loaded in a one-dimensional optical lattice. Furthermore, a
skyrmion-anti-skyrmion lattice is associated with the appearance of such
supersoildity, indicating the topological nontrivial properties of our proposed
supersolids.
