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 $(x_1,\dots ,x_N)$ 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.