Multiple blow-up solutions for the Liouville equation with singular data

2012/10/23 by Teresa D'Aprile, D'Aprile, Teresa · 1 citation
Mathematics · #35B40 #35J20 #35J65 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B40 #msc:35J20 #msc:35J65

paper · pdf · doi:10.48550/arxiv.1210.6270

arxiv created 2012/10/23 · arxiv updated 2012/10/24

Abstract

We study the existence of solutions with multiple concentration to the following boundary value problem -Δu=\e² e^u-4π∑_{p∈ Z}α_p δ_p \hboxin Ω, u=0 \hboxon∂ Ω, where Ω is a smooth and bounded domain in \R², α_p's are positive numbers, Z⊂ Ω is a finite set, δ_p defines the Dirac mass at p, and \e>0 is a small parameter. In particular we extend the result of Del-Pino-Kowalczyk-Musso (\citedelkomu) to the case of several singular sources. More precisely we prove that, under suitable restrictions on the weights α_p, a solution exists with a number of blow-up points ξ_j∈ Ω∖ Z up to ∑_{p∈ Z}max\n∈\N | n<1+α_p\.

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