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| We study the existence of weak solutions to (E) $ (-\Delta)^\alpha
u+g(u)=\nu $ in a bounded regular domain $\Omega$ in $\R^N (N\ge2)$
which vanish in $\R^N\setminus\Omega$, 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 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 $\nu$ 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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