Certain fractional Laplacian equations that do not have smooth solutions

2019/04/11 by José Villa‐Morales, Villa-Morales, José
Computer Science · Mathematics · #35B08 #35J60 #35R11 #60G22 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1904.05975

openalex publication_date 2019/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f be a real-valued function defined on ℝ, with f(0) ≠ 0 and which is not constant in non empty open intervals. We prove the equations \ (-Δ)^su · amp; = · amp; f(u), · amp; in B₁,
u · amp; = · amp; 0, · amp; in B₁^c, . where (-Δ)^s is the s-fractional Laplacian, 0< s <1, have no solutions in C²(B₁), if d>2s. The proof is based on the moving plane method and in the approximation of C²(B₁) functions by s-harmonic functions.

Citations

• Semilinear equations involving the fractional Laplacian on domains
• The Pohozaev Identity for the Fractional Laplacian
• The Brezis-Nirenberg result for the fractional Laplacian
• Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

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• C^1+α regularity for fractional p-harmonic functions