Radial biharmonic k-Hessian equations: The critical dimension

2017/06/18 by Carlos Escudero, Escudero, Carlos, Pedro J. Torres +1
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1706.05684

openalex publication_date 2017/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This work is devoted to the study of radial solutions to the elliptic problem Δ² u = (-1)^k S_k[u] + λf, x ∈ B₁(0) ⊂ ℝ^N, provided either with Dirichlet boundary conditions u = ∂_n u = 0, x ∈ ∂ B₁(0), or Navier boundary conditions u = Δu = 0, x ∈ ∂ B₁(0), where the k-Hessian S_k[u] is the k^th elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum f ∈ L¹(B₁(0)). We also study the existence of entire solutions to this partial differential equation in the case in which they are assumed to decay to zero at infinity and under analogous conditions of summability on the datum. Our results illustrate how, for k=2, the dimension N=4 plays the role of critical dimension separating two different phenomenologies below and above it.

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