### 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 language | English |
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Article number | 99 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 58 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jun 2019 |