Existence, multiplicity and classification results for solutions to k-Hessian equations with general weights

2022/06/23 by João Marcos Ó, Ó, João Marcos do, Justino Sánchez +3
Mathematics · Physics and Astronomy · #34C20 #35J62 #70K05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Nonlinear Waves and Solitons #Primary: 35B33 #Secondary: 34C37

paper · pdf · doi:10.48550/arxiv.2206.11942

openalex publication_date 2022/06/23 · openalex created_date 2022/06/29 · openalex updated_date 2026/08/01

Abstract

The aim of this paper is to study negative classical solutions to a k-Hessian equation involving a nonlinearity with a general weight \begincases S_k(D²u)= λρ(|x|) (1-u)^q amp;in B,
u=0 amp;on ∂ B. \endcases Here, B denotes the unit ball in \mathbb Rⁿ , n>2k, λ is a positive parameter and q>k with k∈ \mathbb N. The function rρ'(r)/ρ(r) satisfies very general conditions in the radial direction r=|x|. We show the existence, nonexistence, and multiplicity of solutions to Problem \eqrefEq:Ma:0. The main technique used for the proofs is a phase-plane analysis related to a non-autonomous dynamical system associated to the equation in \eqrefEq:Ma:0. Further, using the aforementioned non-autonomous system, we give a comprehensive characterization of P₂-, P₃⁺-, P₄⁺-solutions to the related problem \begincases S_k(D² w)= ρ(|x|) (-w)^q,
wlt;0, \endcases given on the entire space \mathbb Rⁿ . In particular, we describe new classes of solutions: fast decay P⁺₃-solutions and P₄⁺-solutions.

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