In this paper, we investigate the qualitative properties of positive solutions for
a two-coupled elliptic system in the punctured space. We establish a monotonicity
formula that completely characterizes the singularity of positive solutions.
We prove a sharp global estimate for both components of positive
solutions. We also prove the nonexistence of positive semi-singular solutions,
which means that one component is bounded near the singularity and the other
component is unbounded near the singularity.
In this paper, we study the asymptotic behavior of positive solutions of fractional
Hardy-Henon equation with an isolated singularity at the origin. We
give a classification of isolated singularities of positive solutions near x = 0. Further,
we prove the asymptotic behavior of positive singular solutions. These
results parallel those known for the Laplacian counterpart proved by Caffarelli, Gidas
and Spruck (Caffarelli, Gidas and Spruck in Comm Pure Appl Math, 1981, 1989),
but the technique is very different, since the ODEs analysis is a missing ingredient in
the fractional case. Our proofs are based on a monotonicity formula, combined with
a blow up (down) argument, the Kelvin transformation and uniqueness of solutions
of related degenerate equations on semi spherical surface. We also investigate
isolated singularities located at infinity of fractional Hardy-Henon equation.