You cannot have a solid phase in two dimensions, as the latter would break translational symmetry and thus violate the Mermin-Wagner theorem.
Rigorous proofs of this fact can be found in the following papers:
Both papers deal with very general interactions (not only Lennard-Jones); the second one even allow for additional hard-core interactions (which are tricky to deal with, from a Mermin-Wagner perspective).
Of course, you might then inquire what ground-states look like, but the latter would not be stable at positive temperatures. Rigorous results about the ground states of systems with a class of interactions resembling Lennard-Jones can be found in the paper
It is proved there that, under suitable assumptions on the boundary conditions, the ground state indeed forms a triangular lattice. Even for ground-states in two dimensions, the problem seems still not fully understood. There is also the following nice (and recent) review paper on this topic: