Liouville-type theorems for the fourth order nonlinear elliptic equation

2013/06/28 by Liang-Gen Hu, Hu, Liang-Gen
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1307.0047

openalex publication_date 2013/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we are concerned with Liouville-type theorems for the nonlinear elliptic equation equation* Δ² u=|x|^a |u|^p-1u in Ω, equation*where a ≥ 0, p>1 and Ω⊂ ℝⁿ is an unbounded domain of ℝⁿ, n ≥ 5. We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the \it Pohozaev-type identity, \it monotonicity formula of solutions and a \it blowing down sequence, which is used to obtain sharper results.

Citations

• Solutions of superlinear elliptic equations and their morse indices
• Global and local behavior of positive solutions of nonlinear elliptic equations

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