Critical conditions and finite energy solutions of several nonlinear elliptic PDEs inRn

Abstract

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.

On critical exponents of ak-Hessian equation in the whole space

“…When p is not larger than the Serrin exponent, (1.1) has no negative kadmissible solution (cf. [15], [22] and [23]). Thus, we always assume in this paper that p is larger than the Serrin exponent p > p se .…”

“…Under the scaling transformation, p = p so ensures that equation (1.1) and energy • p+1 are invariant (cf [15]), and p = p * ensures that equation (1.1) and energy • p+k are invariant (cf [17]). In addition, p * is essential to study the separation property of solutions (see the following Remark).…”

Concentration and energy invariance for a class of fully nonlinear elliptic equations

“…A final remark is that there are other classes of fully nonlinear operators for which there is a clear invariance by rescaling of the related equations and of some associated energy functionals. These are the so called k-Hessian operators which have a variational structure and for which related critical exponents can be defined, sharing many similarities with the case of the Laplacian ( [16], [23]).…”

Some results for semi-stable radial solutions of $k$-Hessian equations

“…Some existence and non-existence results for radial solutions are given in terms of an integral condition involving the function a. In [6], among others results, the author construct explicit negative solutions of the equation F k (D 2 V ) = R(x)(−V ) q in R n , where F k (D 2 V ) = S k (D 2 V ) and R(x) is a radial function that satisfies C −1 ≤ R(x) ≤ C for some constant C > 1. See [6,Theorem 4.2].…”