Fading absorption in non-linear elliptic equations

Abstract

We study the equation $-\Delta u+h(x)|u|^{q-1}u=0$, $q>1$, in $R^N_+=R^{N-1}\ti R_+$ where $h\in C(\bar{R^N_+})$, $h\geq 0$. Let $(x_1,..., x_N)$ be a coordinate system such that $R^N_+=[x_N>0]$ and denote a point $x\in \RN$ by $(x',x_N)$. Assume that $h(x', x_N)>0$ when $x'\neq 0$ but $h(x',x_N)\to 0$ as $|x'|\to 0$. 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.