In this paper, we are concerned with stable solutions to the fractional elliptic equation (-Δ)su=euinRN,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (-\Delta )^s u=e^u \text{ in } {\mathbb {R}}^{N}, \end{aligned}$$\end{document}where (-Δ)s\documentclass[12pt]{minimal}
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\begin{document}$$(-\Delta )^s$$\end{document} is the fractional Laplacian with 0<s<1\documentclass[12pt]{minimal}
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\begin{document}$$0<s<1$$\end{document}. By developing a new technique with non-compactly supported test functions, we establish the nonexistence of stable solutions provided that N<10s\documentclass[12pt]{minimal}
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\begin{document}$$N<10s$$\end{document}. This result is optimal when s↑1\documentclass[12pt]{minimal}
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\begin{document}$$s\uparrow 1$$\end{document}. On the other hand, we believe that our technique can be used to study stable solutions of elliptic equations/systems involving the fractional Laplacian.