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Pré-Publication, Document De Travail Année : 2013

Semilinear fractional elliptic equations involving measures

Huyuan Chen
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Résumé

We study the existence of weak solutions of (E) $ (-\Delta)^\alpha u+g(u)=\nu $ in a bounded regular domain $\Omega$ in $\R^N (N\ge2)$ which vanish on $\R^N\setminus\Omega$, where $(-\Delta)^\alpha$ denotes the fractional Laplacian with $\alpha\in(0,1)$, $\nu$ 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 $\nu$ is Dirac measure, we characterize the asymptotic behavior of the solution. When $g(r)=|r|^{k-1}r$ 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.
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Dates et versions

hal-00820401 , version 1 (04-05-2013)
hal-00820401 , version 2 (14-05-2013)

Identifiants

Citer

Huyuan Chen, Laurent Veron. Semilinear fractional elliptic equations involving measures. 2013. ⟨hal-00820401v2⟩
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