Weak and strong singular solutions of semilinear fractional elliptic equations
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 $(E)_\infty$.
Origine | Fichiers produits par l'(les) auteur(s) |
---|