  We study the continuum epitaxial model for elastic interacting atomic steps
on vicinal surfaces proposed by Xiang and E (Xiang, SIAM J. Appl. Math.
63:241-258, 2002; Xiang and E, Phys. Rev. B 69:035409, 2004). The non-local
term and the singularity complicate the analysis of its PDE. In this paper, we
first generalize this model to the Lennard-Jones (m,n) interaction between
steps. Based on several important formulations of the non-local energy, we
prove the existence, symmetry, unimodality, and regularity of the energy
minimizer in the periodic setting. In particular, the symmetry and unimodality
of the minimizer implies that it has a bunching profile. Furthermore, we derive
the minimum energy scaling law for the original continnum model. All results
are consistent with the corresponding results proved for discrete models by Luo
et al. (Luo et al., Multiscale Model. Simul. 14:737 - 771, 2016).
