  It is well known that elastic effects can cause surface instability. In this
paper, we analyze a one-dimensional discrete system which can reveal pattern
formation mechanism resembling the "step-bunching" phenomena for epitaxial
growth on vicinal surfaces. The surface steps are subject to long-range
pairwise interactions taking the form of a general Lennard--Jones (LJ) type
potential. It is characterized by two exponents $m$ and $n$ describing the
singular and decaying behaviors of the interacting potential at small and large
distances, and henceforth are called generalized LJ $(m,n)$ potential. We
provide a systematic analysis of the asymptotic properties of the step
configurations and the value of the minimum energy, in particular, their
dependence on $m$ and $n$ and an additional parameter $\alpha$ indicating the
interaction range. Our results show that there is a phase transition between
the bunching and non-bunching regimes. Moreover, some of our statements are
applicable for any critical points of the energy, not necessarily minimizers.
This work extends the technique and results of [Luo et al, SIAM MMS, 2016]
which concentrates on the case of LJ (0,2) potential (originated from the
elastic force monopole and dipole interactions between the steps). As a
by-product, our result also leads to the well-known fact that the classical LJ
(6,12) potential does not demonstrate step-bunching type phenomena.
