Fading absorption in non-linear elliptic equations

Marcus, Moshe ; Shishkov, Andrey

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013), p. 315-336 / Harvested from Numdam

Résumé

We study the equation $ -\mathrm{\Delta }u+h(x){|u|}^{q-1}u=0$, $ q>1$, in $ {\mathbb{R}}_{+}^{N}={\mathbb{R}}^{N-1}\times {\mathbb{R}}_{+}$ where $ h\in C(\overline{{\mathbb{R}}_{+}^{N}})$, $ h⩾0$. Let $ ({x}_{1},\dots ,{x}_{N})$ be a coordinate system such that $ {\mathbb{R}}_{+}^{N}=[{x}_{N}>0]$ and denote a point $ x\in {\mathbb{R}}^{N}$ by $ ({x}^{\prime },{x}_{N})$. Assume that $ h({x}^{\prime },{x}_{N})>0$ when $ {x}^{\prime }\ne 0$ but $ h({x}^{\prime },{x}_{N})\to 0$ as $ |{x}^{\prime }|\to 0$. For this class of equations we obtain sharp necessary and sufficient conditions in order that singularities on the boundary do not propagate in the interior.

Détail

Publié le : 2013-01-01

MR 3035979 | Zbl 1295.35208

DOI : https://doi.org/10.1016/j.anihpc.2012.08.002

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