  We propose a minimal interacting lattice model for two-dimensional class-$D$
higher-order topological superconductors with no free-fermion counterpart. A
Lieb-Schultz-Mattis-type constraint is proposed and applied to guide our
lattice model construction. Our model exhibits a trivial product ground state
in the weakly interacting regime, whereas, increasing electron interactions
provoke a novel topological quantum phase transition to a $D_4$-symmetric
higher-order topological superconducting state. The symmetry-protected Majorana
corner modes are numerically confirmed with the matrix-product-state technique.
Our theory paves the way for studying interacting higher-order topology with
explicit lattice model constructions.
