Finite energy solutions and critical conditions of nonlinear equations in $R^n$

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.