Semilinear fractional elliptic equations involving measures |
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Authors: | Huyuan Chen Laurent Véron |
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Institution: | 1. Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi 330022, PR China;2. Departamento de Ingeniería Matemática, CNRS UMR 2071, Universidad de Chile, Santiago, Chile;3. Laboratoire de Mathématiques et Physique Théorique, CNRS UMR 7350, Université François Rabelais, Tours, France |
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Abstract: | We study the existence of weak solutions to (E) (−Δ)αu+g(u)=ν in a bounded regular domain Ω in RN(N≥2) which vanish in RN?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 hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of weak solution for problem (E) for any measure. In the case where ν is a 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. |
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Keywords: | 35R11 35J61 35R06 |
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