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

Let Ω be a bounded domain of R and let uk be solutions to the equation (1) ∆uk = Vke in Ω, where (2) Vk → 1 uniformly in Ω, as k → ∞. Throughout the paper we denote as ∆ = − ∑ i( ∂ ∂xi ) 2 the Laplacian with the geometers’ sign convention. Continuing the analysis of [18], here we study the compactness properties of equation (1). Equation (1) is the fourth order analogue of Liouville’s equation. Thus, for problem (1), (2) we may expect similar results to hold as have been obtained by Brezis-Merle [3] in the two-dimensional case. Recall the following result from [3] (we also refer to Li-Shafrir [11]). Theorem 1.1. Let Σ be a bounded domain of R and let (uk)k∈N be a sequence of solutions to the equation (3) ∆uk = Vke in Σ, where Vk → 1 uniformly in Σ as k →∞, and satisfying the uniform bound

Concentration phenomena for Liouville's equation in dimension four