Boundary singularities of solutions of N-harmonic equations with absorption

TL;DR

This paper investigates the boundary behavior of solutions to N-harmonic equations with absorption, revealing conditions under which solutions are trivial or exhibit weak or strong singularities at the boundary.

Contribution

It characterizes the boundary singularities of solutions to N-harmonic equations with absorption, providing explicit constructions for weak and strong singularities.

Findings

Solutions are identically zero for q ≥ 2N-1.

For (N-1)<q<2N-1, solutions exhibit either weak or strong boundary singularities.

Explicit constructions of boundary singular solutions are provided.

Abstract

We study the boundary behaviour of solutions $u$ of $-\Delta_{N}u+ |u|^{q-1}u=0$ in a bounded smooth domain $\Omega\subset\mathbb R^{N}$ subject to the boundary condition $u=0$ except at one point, in the range $q>N-1$. We prove that if $q\geq 2N-1$ such a $u$ is identically zero, while, if $N-1<q<2N-1$, $u$ inherits a boundary behaviour which either corresponds to a weak singularity, or to a strong singularity. Such singularities are effectively constructed.