Multiplicity of bounded solutions to the k-Hessian equation with a Matukuma-type source

2018/07/31 by Yasuhito Miyamoto, Miyamoto, Yasuhito, Justino Sánchez +3
Mathematics · Physics and Astronomy · #34C20 #35J62 #70K05 #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Nonlinear Waves and Solitons #primary 35B33 #secondary 34C37

paper · pdf · doi:10.48550/arxiv.1807.11644

openalex publication_date 2018/07/31 · openalex created_date 2018/08/22 · openalex updated_date 2026/07/28

Abstract

The aim of this paper is to deal with the k-Hessian counterpart of the Laplace equation involving a nonlinearity studied by Matukuma. Namely, our model is the problem (1) \begincases S_k(D²u)= λ\frac|x|^μ-2(1+|x|²)^\fracμ2 (1-u)^q amp;in B,
u lt;0 amp; in B,
u=0 amp;on ∂ B, \endcases where B denotes the unit ball in ℝⁿ,n>2k (k∈ℕ), λ>0 is an additional parameter, q>k and μ≥ 2. In this setting, through a transformation recently introduced by two of the authors that reduces problem (1) to a non-autonomous two-dimensional generalized Lotka-Volterra system, we prove the existence and multiplicity of solutions for the above problem combining dynamical-systems tools, the intersection number between a regular and a singular solution and the super and subsolution method.

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

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