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The article is an attempt to investigate the issues of asymptotic analysisfor problems involving fractional Laplacian where the domains tend to becomeunbounded in one-direction. Motivated from the pioneering work on second orderelliptic problems by Chipot and Rougirel, where the force functions areconsidered on the cross section of domains, we prove the non-local counterpartof their result.  Furthermore, recently Yeressian established a weighted estimate for solutionsof nonlocal Dirichlet problems which exhibit the asymptotic behavior. The casewhens= 1=2 was also treated as an example to show how the weighted estimatemight be used to achieve the asymptotic behavior. In this article, we extendthis result to each order between 0 and 1.

On the Asymptotic Analysis of Problems Involving Fractional Laplacian in Cylindrical Domains Tending to Infinity