  We prove that any positive solution of the Yamabe equation on an
asymptotically flat $n$-dimensional manifold of flatness order at least
$\frac{n-2}{2}$ and $n\le 24$ must converge at infinity either to a fundamental
solution of the Laplace operator on the Euclidean space or to a radial Fowler
solution defined on the entire Euclidean space. The flatness order
$\frac{n-2}{2}$ is the minimal flatness order required to define ADM mass in
general relativity; the dimension $24$ is the dividing dimension of the
validity of compactness of solutions to the Yamabe problem. We also prove such
alternatives for bounded solutions when $n>24$.
  We prove these results by establishing appropriate asymptotic behavior near
an isolated singularity of solutions to the Yamabe equation when the metric has
a flatness order of at least $\frac{n-2}{2}$ at the singularity and $n\le 24$,
also when $n>24$ and the solution grows no faster than the fundamental solution
of the flat metric Laplacian at the singularity. These results extend earlier
results of L. Caffarelli, B. Gidas and J. Spruck, also of N. Korevaar, R.
Mazzeo, F. Pacard and R. Schoen, when the metric is conformally flat, and work
of C.C. Chen and C. S. Lin when the scalar curvature is a non-constant function
with appropriate flatness at the singular point, also work of F. Marques when
the metric is not necessarily conformally flat but smooth, and the dimension of
the manifold is three, four, or five, as well as recent similar results by the
second and third authors in dimension six.
