Classification of positive solutions to [formula omitted]-Laplace equations with critical Hardy–Sobolev exponent

Abstract

Let u∈D1,p(RN) be a positive solution to the critical equation −Δpu=upa∗−1|x|ain RN,where 1<p<N, 0<a<p and pa∗=(N−a)pN−p. By exploiting the method of moving planes, we show that u is radial and strictly radially decreasing about the origin. Consequently, u must have the form u(x)=(N−a)1pN−pp−1p−1pλλp+|x|p−ap−1N−pp−afor some λ>0.