Uniqueness of positive solutions to the higher order Brezis-Nirenberg problem

2022/10/13 by Zhongwei Tang, Tang, Zhongwei, Zhou, Ning
Computer Science · Mathematics · #35A02 #35J08 #35J30 #35J91 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2210.06793

openalex publication_date 2022/10/13 · openalex created_date 2022/10/15 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the higher order Brezis-Nirenberg problem under the Navier boundary condition \be \begincases (-Δ)^m u=ε u+u^p & \text in Ω, u>0 & \text in Ω, u=-Δu=⋯=(-Δ)^m-1 u=0 & \text on ∂ Ω, \endcases \ee where Ω is a strictly convex smooth bounded domain in ℝⁿ with n ≥ 4m, m ∈ ℕ₊, ε∈ (0,λ₁), λ₁ is the first Navier eigenvalue for (-Δ)^m in Ω, and p=(n+2m)/(n-2m). We prove that the solutions of \eqrefeq are unique if either ε close to λ₁ or ε close to 0 and Ω satisfies some symmetry assumptions. The proof is mainly based on our previous works about the blow up analysis and compactness result for solutions to higher order critical elliptic equations and the asymptotic behavior of solutions to \eqrefeq.

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