Radially bounded solutions of a k-Hessian equation involving a weighted nonlinear source

2016/03/23 by Justino Sánchez, Justino Sanchez, Sanchez, Justino +2
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1603.07280

arXiv admin note: substantial text overlap with arXiv:1510.07669

arxiv created 2016/03/23 · openalex publication_date 2016/03/23 · arxiv updated 2016/03/24 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We consider the problem (1) \begincases S_k(D²u)= λ|x|^σ (1-u)^q in B,
u <0 in B,
u=0 on ∂ B, \endcases where B denotes the unit ball in ℝⁿ, n>2k (k∈ ℕ), λ>0, q > k and σ≥ 0. We study the existence, uniqueness and multiplicity of negative bounded radially symmetric solutions of (1). The methodology to obtain our results is based on a dynamical system approach. For this, we introduce a new transformation which reduces problem (1) to an autonomous two dimensional generalized Lotka-Volterra system.

Citations

• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian
• Quasilinear Dirichlet problems driven by positive sources
• Bounded solutions of a k-Hessian equation in a ball

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