Semilinear fractional elliptic equations involving measures

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:257

Article

Chen, Huyuan^1,2  Veron, Laurent³ 

[1] Jiangxi Normal Univ, Dept Math, Nanchang 330022, Jiangxi, Peoples R China

[2] Univ Chile, Dept Ingn Matemat, CNRS UMR 2071, Santiago, Chile

[3] Univ Tours, CNRS UMR 7350, Lab Math & Phys Theor, Tours, France

关键词: Fractional Laplacian;    Radon measure;    Dirac measure;    Green kernel;    Bessel capacities;   

DOI  :  10.1016/j.jde.2014.05.012

来源: Elsevier

【 摘 要 】

We study the existence of weak solutions to (E) (-Delta)(alpha)u g(u) = v in a bounded regular domain ohm in R-N (N >= 2) which vanish in R-N \ ohm, where (-Delta)(alpha) denotes the fractional Laplacian with alpha is an element of (0, 1), v 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 v is a Dirac measure, we characterize the asymptotic behavior of the solution. When g(r) = vertical bar r vertical bar(k-1) r 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. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free