Fading absorption in non-linear elliptic equations

Marcus, Moshe  ; Shishkov, Andrey 

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 2, pp. 315-336

Résumé

We study the equation $-\Delta u+h\left(x\right){\left|u\right|}^{q-1}u=0$, $q>1$, in ${ℝ}_{+}^N={ℝ}^{N-1}\times {ℝ}_{+}$ where $h\in C\left(\overline{{ℝ}_{+}^N}\right)$, $h⩾0$. Let $(x_1,\cdots ,x_N)$ be a coordinate system such that ${ℝ}_{+}^N=[x_N>0]$ and denote a point $x\in {ℝ}^N$ by $(x^{\text{'}},x_N)$. Assume that $h(x^{\text{'}},x_N)>0$ when $x^{\text{'}}\ne 0$ but $h(x^{\text{'}},x_N)\to 0$ 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.

Détail

DOI : 10.1016/j.anihpc.2012.08.002