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 $${\left( { - \Delta } \right)^\sigma }u = K\left( x \right){u^{{{n + 2\sigma \over {n }}\,}}\,\,\,{\rm{in}}\,{B_2}\backslash \left\{ 0 \right\}$$ with an isolated singularity at origin, where K is a function on B2, punctured ball B2 {0} ⊂ ℝn n ⩾ 2, σ ∈ (0, 1), and (−Δ)σ Laplacian. In lower dimensions, we show that for any C1(B2), solution u always satisfies u(x) ⩽ C∣x∣−(n−2σ)/2 near origin. contrast, construct functions C1(B2) in higher dimensions such could be arbitrarily large particular, these results also apply prescribed boundary mean curvature equations $${\mathbb{B}^{n 1}}$$ .