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It is well known that elastic effects can cause surface instability. In thispaper, we analyze a one-dimensional discrete system which can reveal patternformation mechanism resembling the "step-bunching" phenomena for epitaxialgrowth on vicinal surfaces. The surface steps are subject to long-rangepairwise interactions taking the form of a general Lennard--Jones (LJ) typepotential. It is characterized by two exponents $m$ and $n$ describing thesingular and decaying behaviors of the interacting potential at small and largedistances, and henceforth are called generalized LJ $(m,n)$ potential. Weprovide a systematic analysis of the asymptotic properties of the stepconfigurations and the value of the minimum energy, in particular, theirdependence on $m$ and $n$ and an additional parameter $\alpha$ indicating theinteraction range. Our results show that there is a phase transition betweenthe bunching and non-bunching regimes. Moreover, some of our statements areapplicable 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 theelastic force monopole and dipole interactions between the steps). As aby-product, our result also leads to the well-known fact that the classical LJ(6,12) potential does not demonstrate step-bunching type phenomena.

Energy Scaling and Asymptotic Properties of One-Dimensional Discrete System with Generalized Lennard--Jones $(m,n)$ Interaction