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Pré-Publication, Document De Travail Année : 2013

Weak and strong singular solutions of semilinear fractional elliptic equations

Huyuan Chen
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Résumé

If $p\in(0,\frac{N}{N-2\alpha})$, $\alpha\in(0,1)$, $k>0$ and $\Omega\subset \R^N$ is a bounded $C^2$ domain containing $0$ and $\delta_0$ is the Dirac measure at $0$, we prove that the weak solution of $(E)_k$ $ (-\Delta)^\alpha u+u^p=k\delta_0 $ in $\Omega$ which vanishes in $\Omega^c$ is a weak singular solution of $(E)_\infty$ $ (-\Delta)^\alpha u+u^p=0 $ in $\Omega\setminus\{0\}$ with the same outer data. Furthermore, we study the limit of weak solutions of $(E)_k$ when $k\to\infty$. For $p\in(0, 1+\frac{2\alpha}{N}]$, the limit is infinity in $\Omega$. For $p\in(1+\frac{2\alpha}N,\frac{N}{N-2\alpha})$, the limit is a strong singular solution of $(E)_\infty$.
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Dates et versions

hal-00848582 , version 1 (26-07-2013)
hal-00848582 , version 2 (17-11-2013)
hal-00848582 , version 3 (26-11-2013)

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Huyuan Chen, Laurent Veron. Weak and strong singular solutions of semilinear fractional elliptic equations. 2013. ⟨hal-00848582v1⟩
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