In this paper, we prove the Pohozaev identity for the following fractional nonlinear elliptic equation: \(\begin{aligned} (-\Delta )^{s}u= g(u) \text{ in } {\mathbb {R}}^{N}, \end{aligned}\) where \(N\ge 2\) , \(s\in (0, 1)\) , \((-\Delta )^{s}\) denotes the fractional Laplacian operator, and \(g:{\mathbb {R}}\rightarrow {\mathbb {R}}\) is a locally Hölder continuous function. Our proof rests on a regularity result for bounded distributional solutions of the above equation, an integration by parts formula for \({\mathcal {D}}^{s, 2}\cap C^{2s+{{\,\mathrm{\varepsilon }\,}}}\) functions with locally bounded gradient and vector fields of class \(C^{0, 1}_{c}\) , and a limiting procedure.
