Semilinear fractional elliptic equations involving measures

2013/05/04 by Huyuan Chen, Chen, Huyuan, Лаурент Верон +1 · 2 citations
Computer Science · Mathematics · #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.1305.0945

openalex publication_date 2013/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the existence of weak solutions of (E) (-Δ)^αu+g(u)=ν in a bounded regular domain Ω in \R^N (N≥2) which vanish on \R^N∖Ω, where (-Δ)^α denotes the fractional Laplacian with α∈(0,1), ν is a Radon measure and g is a nondecreasing function satisfying some extra hypothesis. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of a weak solution for problem (E) for any measure. In the case where ν is Dirac measure, we characterize the asymptotic behavior of the solution. When g(r)=|r|^k-1r with k supercritical, we show that a condition of absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved.

Citations

• Singular solutions of fractional elliptic equations with absorption
• Classification of solutions for an integral equation
• Regularity of the obstacle problem for a fractional power of the laplace operator
• Hitchhiker's guide to the fractional Sobolev spaces
• Quasilinear Lane-Emden equations with absorption and measure data
• The Dirichlet problem for the fractional Laplacian: regularity up to the\n boundary
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Cited by

• Isolated Singularities and Measure Data Problems for Semilinear Equations Driven by Stable Lévy Operators
• Large solutions to semilinear equations for subordinate Laplacians in C^1,1 bounded open sets

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