  This paper is concerned with the critical conditions of nonlinear elliptic
equations with weights and the corresponding integral equations with Riesz
potentials and Bessel potentials. We show that the equations and some energy
functionals are invariant under the scaling transformation if and only if the
critical conditions hold. In addition, the Pohozaev identity shows that those
critical conditions are the necessary and sufficient conditions for existence
of the finite energy positive solutions or weak solutions. Finally, we discuss
respectively the existence of the negative solutions of the $k$-Hessian
equations in the subcritical case, critical case and supercritical case. Here
the Serrin exponent and the critical exponent play key roles.
