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

Abstract

We study positive solutions to the fractional semi-linear elliptic equation $${\left( { - \Delta } \right)^\sigma }u = K\left( x \right){u^{{{n + 2\sigma } \over {n - 2\sigma }}\,}}\,\,\,{\rm{in}}\,{B_2}\backslash \left\{ 0 \right\}$$ with an isolated singularity at the origin, where K is a positive function on B2, the punctured ball B2 {0} ⊂ ℝn with n ⩾ 2, σ ∈ (0, 1), and (−Δ)σ is the fractional Laplacian. In lower dimensions, we show that for any K ∈ C1(B2), a positive solution u always satisfies that u(x) ⩽ C∣x∣−(n−2σ)/2 near the origin. In contrast, we construct positive functions K ∈ C1(B2) 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 $${\mathbb{B}^{n + 1}}$$ .