Fading absorption in non-linear elliptic equations
Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 2, pp. 315-336.

We study the equation $-\Delta u+h\left(x\right){|u|}^{q-1}u=0$, $q>1$, in ${ℝ}_{+}^{N}={ℝ}^{N-1}×{ℝ}_{+}$ where $h\in C\left(\overline{{ℝ}_{+}^{N}}\right)$, $h⩾0$. Let $\left({x}_{1},\cdots ,{x}_{N}\right)$ be a coordinate system such that ${ℝ}_{+}^{N}=\left[{x}_{N}>0\right]$ and denote a point $x\in {ℝ}^{N}$ by $\left({x}^{\text{'}},{x}_{N}\right)$. Assume that $h\left({x}^{\text{'}},{x}_{N}\right)>0$ when ${x}^{\text{'}}\ne 0$ but $h\left({x}^{\text{'}},{x}_{N}\right)\to 0$ as $|{x}^{\text{'}}|\to 0$. 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.

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author = {Marcus, Moshe and Shishkov, Andrey},
title = {Fading absorption in non-linear elliptic equations},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {315--336},
publisher = {Elsevier},
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Marcus, Moshe; Shishkov, Andrey. Fading absorption in non-linear elliptic equations. Annales de l'I.H.P. Analyse non linéaire, Volume 30 (2013) no. 2, pp. 315-336. doi : 10.1016/j.anihpc.2012.08.002. http://www.numdam.org/articles/10.1016/j.anihpc.2012.08.002/

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