A Liouville type result for fractional GJMS equations on higher dimensional spheres

2023/05/12 by Quỳnh N. T. Lê, Lê, Quynh N. T., Quốc Anh Ngô +3
Mathematics · #35A15 #35A23 #35C15 #45H05 #58J70 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2305.07249

openalex publication_date 2023/05/12 · openalex created_date 2023/05/17 · openalex updated_date 2026/07/28

Abstract

Let n be an integer and s be a real number such that n > 2s ≥ 2. Inspired by the perturbation approach initiated by F. Hang and P. Yang (Int. Math. Res. Not. IMRN, 2020), we are interested in non-negative, smooth solution v to the following higher-order fractional equation \mathbf P_n^2s(v) = Q_n^2s(ε v+v^α) on \mathbf Sⁿ with 0<α≤ (n+2s)/(n-2s), and ε ≥ 0. Here \mathbf P_n^2s is the fractional GJMS type operator of order 2s on \mathbf Sⁿ and Q_n^2s =\mathbf P_n^2s(1) is constant. We show that if ε >0 and 0<α≤ (n+2s)/(n-2s), then any positive, smooth solution v to the above equation must be constant. The same result remains valid if ε=0 but with 0<α< (n+2s)/(n-2s).As a by-product, with 0<α≤ (n+2s)/(n-2s), we compute the sharp constant of the subcritical/critical Sobolev inequalities ∫_{\mathbf Sⁿ} v \mathbf P_n^2s (v) dμ_g_{\mathbf Sⁿ} ≥ (Γ(n/2 + s))/(Γ(n/2 - s )) | \mathbf Sⁿ|^(α-1)/(α+1) ( ∫_{\mathbf Sⁿ} v^α+1 dμ_g_{\mathbf Sⁿ} )^(2)/(α+1). for the GJMS operator \mathbf P_n^2s on \mathbf Sⁿ and for all non-negative functions v∈ H^s(\mathbf Sⁿ).

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