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

On isolated singularities for fractional Lane-Emden equation in the Serrin's critical case