Existence of bubbling solutions for the Liouville system in a torus

Hsin-Yuan Huang*

*Corresponding author for this work

Research output: Contribution to journalArticle

Abstract

We consider the following Liouville system on a parallelogram Ω in R 2 : Δui+∑j=1naijρj(hjeuj∫Ωhjeuj-1|Ω|)=0,i∈I={1,…,n},where h i (x) ∈ C 3 (Ω) , h i (x) > 0 , u i is doubly periodic on ∂Ω (i∈ I), and A=(aij)n×n is a non-negative constant matrix. We prove that if q is a non-degenerate critical point of ∑i=1nρi∗loghi(x) and A satisfies certain conditions stated in Theorem 1.1, (0.1) has a sequence of fully bubbling solutions which blow up at p, as ρ=(ρ1,…,ρn)→ρ∗=(ρ1∗,…,ρn∗), where ρ satisfies 8π∑i=1nρi∗=∑i=1n∑j=1naijρi∗ρj∗ and ∑i=1naijρi∗ρj∗>6π for j∈ I.

Original languageEnglish
Article number99
JournalCalculus of Variations and Partial Differential Equations
Volume58
Issue number3
DOIs
StatePublished - 1 Jun 2019

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