Boundary singularities of solutions of N-harmonic equations with absorption

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 $q\ge 2N-1$ 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.