On the prescribed scalar curvature problem with very degenerate prescribed functions

We revisit the well known prescribed scalar curvature problem{-delta u=(1+epsilon K(x))u2 & lowast;-1,u(x)> 0, u is an element of D1,2(RN),x is an element of RN,{-delta u=(1+epsilon K(x))u2 & lowast;-1,u(x)> 0, x is an element of RN,u is an element of D-1,D-2(RN),where 2 & lowast;=2NN-22 & lowast;=2NN-2, N >= 5N >= 5, epsilon > 0 epsilon > 0 and K(x)is an element of C1(RN)& cap;L infinity(RN)K(x)is an element of C1(RN)& cap;L infinity(RN). It is known that there are a number of results related to the existence of solutions concentrating at the isolated critical points of K(x). However, if K(x) has non-isolated critical points with different degenerate rates along different directions, whether there exist solutions concentrating at these points is still an open problem. We give a certain positive answer to this problem via applying a blow-up argument based on local Pohozaev identities and modified finite dimensional reduction method when the dimension of critical point set of K(x) ranges from 1 to N-1N-1, which generalizes some results in Cao et al. (Calc Var Partial Differ Equ 15:403-419, 2002) and Li (J Differ Equ 120:319-410, 1995; Commun Pure Appl Math 49:541-597, 1996).