Semilinear fractional elliptic equations involving measures

By Huyuan Chen, Laurent Véron. 2013

129 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/4f2e2b04472fd8296d7b484cf4739829809b864f

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Citations in the corpus

Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian (2013, background): “[8] Huyan Chen and Laurent Véron.”
Nonlocal Equations with Measure Data (2014, background): “More recent work in [17] deals again with fractional equations involving measures, this time for any s > 0, while the operator is given by the fractional Laplacean, and the analysis is carried out by means of fundamental solutions.”