Fading absorption in non-linear elliptic equations

TL;DR

This paper investigates conditions under which boundary singularities in a class of non-linear elliptic equations do not extend into the interior, focusing on the role of the coefficient function's behavior near the boundary.

Contribution

It provides sharp necessary and sufficient conditions for the non-propagation of boundary singularities in non-linear elliptic equations with variable coefficients.

Findings

Identifies conditions on h(x',x_N) for singularity non-propagation

Shows that h(x',x_N) approaching zero influences singularity behavior

Establishes criteria for boundary singularity control in 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.