  We study the 2+1 dimensional continuum model for long-range elastic
interaction on stepped epitaxial surface proposed by Xu and Xiang. The
long-range interaction term and the two length scales in this model present
challenges for the PDE analysis. In this paper, we prove the existence of both
the static and dynamic solutions and derive the minimum energy scaling law for
this 2+1 dimensional continuum model. We show that the minimum energy surface
profile is attained by surfaces with step meandering instability, which is
essentially different from the energy scaling law for the 1+1 dimensional
epitaxial surfaces under elastic effects, which is attained by step bunching
surface profiles. We also discuss the transition from the step bunching
instability to the step meandering instability in 2+1 dimensions.
