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This paper is concerned with the critical conditions of nonlinear elliptic equations and the corresponding integral equations involving 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. 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 type critical exponent and the Sobolev type critical exponent play key roles. [ABSTRACT FROM AUTHOR]

In an earlier paper of the author''s, partial differential equations with constant coefficients have been studied. Under a certain (restrictive) assumption upon the equation, those initial conditions were characterized for which the normalized formal solution of a corresponding Cauchy problem is k-summable. Here we treat the general situation and prove an analogous result, using multisummability instead of k-summability. The appropriate multisummability type is shown to depend upon the given PDE only, and can be determined from a corresponding Newton polygon. [Copyright &y& Elsevier]