Fading absorption in non-linear elliptic equations

Abstract

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(\mathbb{R}_{ + }^{N}\limits^{¯}), $h⩾0$. Let $(x_{1},…,x_{N})$ be a coordinate system such that $\mathbb{R}_{ + }^{N} = [x_{N} > 0]$ and denote a point $x \in \mathbb{R}^{N}$ by $(x^{′},x_{N})$. Assume that $h(x^{′},x_{N}) > 0$ when $x^{′} \neq 0$ but $h(x^{′},x_{N})\rightarrow 0$ as $|x^{′}|\rightarrow 0$. 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.