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

作　　者：Xusheng Du Hui Yang 

机构地区：[1]Department of Mathematics,The Hong Kong University of Science and Technology,Hong Kong,China

出　　处：《Science China Mathematics》2023年第9期1965-1992,共28页中国科学：数学（英文版）

摘　　要：We study positive solutions to the fractional semi-linear elliptic equation(−∆)σu=K(x)u n+2σn−2σin B2\{0}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(−∆)σis the fractional Laplacian.In lower dimensions,we show that for any K∈C1(B2),a positive solution u always satisfies that u(x)6 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 B n+1.

关 键 词：fractional elliptic equations boundary mean curvature equations local estimates large singular solutions 

分 类 号：O175.25[理学—数学]