  In this paper, we give a full classification of the nonexistence of positive
weak solutions to the semilinear elliptic inequality involving the fractional
Hardy potential in punctured and in exterior domains. Our methods are
self-contained and new. The main ideas and key ingredients will be discussed in
the next section after Theorem 1.1 for punctured domains, and after Theorem 1.4
for exterior domains. We will also explain why all the previous methods and
techniques do not apply to our general setting. Let us give here a foretaste of
our line of attack: Based on the imbalance between the Hardy operator and the
nonlinearity, we can obtain an initial asymptotic behavior rate at the origin (
for punctured domains) or at infinity ( exterior domains).We then improve this
rate by using the interaction with the nonlinearity. By repeating this process
finite number of times, a contradiction will be deduced from the nonexistence
for the related non-homogeneous fractional Hardy problem. This process allows
us to obtain the nonexistence for the fractional Hardy problem with larger
ranges . Our study covers all possible ranges, and our results are optimal.
