  In this paper, we concern the isolated singular solutions for semi-linear
elliptic equations involving the Hardy-Leray potentials
  \begin{equation}\label{0}
  -\Delta u+\frac{\mu}{|x|^2} u=u^p\quad {\rm in}\quad
\Omega\setminus\{0\},\qquad u=0\quad{\rm on}\quad \partial\Omega.
  \end{equation}
  We classify the isolated singularities and obtain the existence, the
stability of positive solutions of (\ref{0}). Our results are based on the
study of nonhomogeneous Hardy problem in a new distributional sense.
