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

@inproceedings{Du2021APE, title={A priori estimates versus arbitrarily large solutions for fractional semi-linear elliptic equations with critical Sobolev exponent}, author={Xusheng Du and Hui Yang}, year={2021} }

We study positive solutions to the fractional semi-linear elliptic equation (−∆)u = K(x)u n+2σ n−2σ in B2 \ {0} with an isolated singularity at the origin, where K is a positive function on B2, the punctured ball B2 \ {0} ⊂ R n with n ≥ 2, σ ∈ (0, 1), and (−∆)σ is the fractional Laplacian. In lower dimensions, we show that, for any K ∈ C(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 ∈ C(B2) in higher…

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