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

Abstract

AbstractIn this paper, we study negative classical solutions and stable solutions of the followingk-Hessian equation$$F_k(D^2V) = (-V)^p\quad {\rm in}\;\; R^n$$with radial structure, wheren⩾ 3, 1 <k<n/2 andp> 1. This equation is related to the extremal functions of the Hessian Sobolev inequality on the whole space. Several critical exponents including the Serrin type, the Sobolev type, and the Joseph-Lundgren type, play key roles in studying existence and decay rates. We believe that these critical exponents still come into play to researchk-Hessian equations without radial structure.