  The article is an attempt to investigate the issues of asymptotic analysis
for problems involving fractional Laplacian where the domains tend to become
unbounded in one-direction. Motivated from the pioneering work on second order
elliptic problems by Chipot and Rougirel, where the force functions are
considered on the cross section of domains, we prove the non-local counterpart
of their result.
  Furthermore, recently Yeressian established a weighted estimate for solutions
of nonlocal Dirichlet problems which exhibit the asymptotic behavior. The case
whens= 1=2 was also treated as an example to show how the weighted estimate
might be used to achieve the asymptotic behavior. In this article, we extend
this result to each order between 0 and 1.
