  In this paper, we solve the fractional Lane-Emden equation in the Serrin's
critical case for the fractional Laplacian by developing an innovative and
self-contained approach that also applies to the classical setting (
Laplacian). We give a classification of the isolated singularities of positive
solutions to the semilinear fractional elliptic equations $$(E) \qquad\qquad
(-\Delta)^s u = u^{\frac{N}{N-2s}}\quad {\rm in}\ \ \Omega\setminus\{0\},\qquad
u\geq 0\quad{\rm in}\ \ \mathbb{R}^N\setminus\Omega,\qquad\qquad\quad $$ where
$s\in(0,1)$, $\Omega$ is a bounded domain containing the origin in
$\mathbb{R}^N$ with $N>2s$ and $\frac{N}{N-2s}$ is the Serrin's critical
exponent. We use an initial asymptotic at infinity to transform the critical
case into a subcritical case where the underlying equation involves the
fractional Hardy operator. The construction of singular solutions is based on
the fact that some special functions are subsolutions of the original problem
near the origin.
  We also classify the non-removable singularities of the solutions of $(E)$
and show the existence of a sequence of isolated singular solutions
parameterizing the coefficients of the second order blow up term. To the best
of our knowledge, this idea has also been first used in this work.
