Abstract

The present paper is concerned with negative classical solutions to a k-Hessian equation involving a nonlinearity with a general weight {Sk(D2u)=??(|x|)(1?u)qin B,u=0on ?B. Here, B denotes the unit ball in Rn, n>2k, ? is a positive parameter and q>k with k?N. The function r??(r)/?(r) satisfies very general conditions in the radial direction r=|x|. We show the existence, nonexistence, and multiplicity of solutions to Problem (P). The main technique used for the proofs is a phase-plane analysis related to a non-autonomous dynamical system associated to the equation in (P). Further, using the aforementioned non-autonomous system, we give a comprehensive characterization of P2-, P3+-, P4+-solutions to the related problem (Pˆ){Sk(D2w)=?(|x|)(?w)q,w<0, given on the entire space Rn. In particular, we describe new classes of solutions: fast decay P3+-solutions and P4+-solutions. © 2025 Elsevier Inc.