Compactness of extremals for critical singular Trudinger-Moser functionals

Abstract

Let Ω be a smooth bounded domain in R2, and W01,2(Ω) be the usual Sobolev space. Assume 0∈Ω. For any 0<β<1, according to Csato-Roy [7] and Yang-Zhu [21], there exists some function uβ∈W01,2(Ω)∩Cloc1(Ω‾∖{0})∩C0(Ω‾) satisfying∫Ωe4π(1−β)uβ2|x|2βdx=supu∈W01,2(Ω),‖u‖W01,2(Ω)≤1⁡∫Ωe4π(1−β)u2|x|2βdx. In this paper, we concern the compactness of the function family (uβ)0<β<1. It is shown that up to a subsequence, uβ converges to some function u0 in C1(Ω‾) as β tends to 0. Furthermore, u0 is an extremal of the supremumsupu∈W01,2(Ω),‖u‖W01,2(Ω)≤1⁡∫Ωe4πu2dx.The geometric meaning reads as follows: If we write g0=dx12+dx22 and gβ=|x|−2β(dx12+dx22) as the Euclidean and conical metrics on Ω respectively, where x=(x1,x2)∈Ω, then as β→0, the extremal family (uβ)0<β<1 of the singular Trudinger-Moser functionalsJβ(u)=∫Ωe4π(1−β)u2dvgβ is compact under the constraint u∈W01,2(Ω,gβ) and ‖u‖W01,2(Ω,gβ)≤1.