Abstract: In this talk we will talk about how to determine  the exact number  of positive solutions or positive blow-up solutions to elliptic problems with critical exponent. The main arguments are  based on the local Pohozaev identities, blow-up analysis and the properties on the Green's function. This talk  is based on  joint papers with Prof. Daomin Cao、Shusen Yan and Dr. Peng Luo, etc.