English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 isdominated by a harmonic function and $\sigma$-moderate if it is the limit of anincreasing sequence of moderate solutions. It is known that in the sub-criticalcase, $1<q<q_c=(N+1)/(N-1)$, every positive solution is $\sigma$-moderate [31].More recently Dynkin proved, by probabilistic methods, that this remains validin 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 solutionis $\sigma$-moderate. We use purely analytic techniques which apply to the fullsuper-critical range. The main tools come from linear and non-linear potentialtheory. Combined with previous results, this establishes a 1-1 correspondencebetween positive solutions and their boundary traces in the sense of [35].

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