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

We study positive solutions to the fractional semi-linear elliptic equation $$ (- Δ)^σu = K(x) u^\frac{n + 2 σ}{n - 2 σ} ~~~~~~ in ~ B_2 \setminus \{ 0 \} $$ 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 $n \geq 2$, $σ\in (0, 1)$, and $(- Δ)^σ$ 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 $u(x) \leq C |x|^{ - (n - 2 σ)/2 }$ near the origin. In contrast, we construct positive functions $K \in C^1 (B_2)$ in higher dimensions such that a positive solution $u$ could be arbitrarily large near the origin. In particular, these results also apply to the prescribed boundary mean curvature equations on $\mathbb{B}^{n+1}$.