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We study positive solutions to the fractional semi-linear elliptic equation$$ (- \Delta)^\sigma u = K(x) u^\frac{n + 2 \sigma}{n - 2 \sigma}  in ~B_2 \setminus \{ 0 \} $$ with an isolated singularity at the origin, where $K$is a positive function on $B_2$, the punctured ball $B_2 \setminus \{ 0 \}\subset \mathbb{R}^n$ with $n \geq 2$, $\sigma \in (0, 1)$, and $(-\Delta)^\sigma$ is the fractional Laplacian. In lower dimensions, we show that,for any $K \in C^1 (B_2)$, a positive solution $u$ always satisfies that $u(x)\leq C |x|^{ - (n - 2 \sigma)/2 }$ near the origin. In contrast, we constructpositive functions $K \in C^1 (B_2)$ in higher dimensions such that a positivesolution $u$ could be arbitrarily large near the origin. In particular, theseresults also apply to the prescribed boundary mean curvature equations on$\mathbb{B}^{n+1}$.

A priori estimates versus arbitrarily large solutions for fractional semi-linear elliptic equations with critical Sobolev exponent