Complete classification of the positive solutions of $-\Delta u + u^q=0$

TL;DR

This paper proves that all positive solutions of the nonlinear PDE $- riangle u + u^q=0$ in a bounded domain are $\sigma$-moderate for all $q eq 1$, extending previous results to the super-critical range using analytic potential theory.

Contribution

It establishes that every positive solution is $\sigma$-moderate for all $q eq 1$, including the super-critical case, using purely analytic methods.

Findings

All positive solutions are $\sigma$-moderate for $q eq 1$.

The result applies to the full super-critical range $q eq 1$.

Establishes a 1-1 correspondence between solutions and boundary traces.

Abstract

We study the equation $-\Delta u+u^q=0$, $q>1$, in a bounded $C^2$ domain $\Omega\subset R^N$. A positive solution of the equation is moderate if it is dominated by a harmonic function and $\sigma$-moderate if it is the limit of an increasing sequence of moderate solutions. It is known that in the sub-critical case, $1<q<q_c=(N+1)/(N-1)$, every positive solution is $\sigma$-moderate [31]. More recently Dynkin proved, by probabilistic methods, that this remains valid in the super-critical case for $q\le2$, [15]. The question remained open for $q>2$. In this paper we prove that, for all $q\ge q_c$, every positive solution is $\sigma$-moderate. We use purely analytic techniques which apply to the full super-critical range. The main tools come from linear and non-linear potential theory. Combined with previous results, this establishes a 1-1 correspondence between positive solutions and their boundary traces in the sense of [35].