We will talk about qualitative properties of positive solutions for some second and fourth order elliptic equations. In particular we discuss the stability for solutions of biharmonic equations with negative exponent and the Henon type equation with exponential nonlinearity respectively. We emphasize the role plays by the exponent of the nonlinearity for the existence and/or nonexistence. This is based on joint works with X. Huang, Z.M. Guo, and D. Ye.