TY - JOUR

T1 - The geometry of generalized Lamé equation, II

T2 - Existence of pre-modular forms and application

AU - Chen, Zhijie

AU - Kuo, Ting Jung

AU - Lin, Chang Shou

N1 - Funding Information:
The authors thank the anonymous referee very much for valuable comments. The research of the first author was supported by NSFC (Grant No. 11701312 , 11871123 ).
Publisher Copyright:
© 2019 Elsevier Masson SAS

PY - 2019/12

Y1 - 2019/12

N2 - In this paper, the second in a series, we continue to study the generalized Lamé equation with the Treibich-Verdier potential y″(z)=[∑k=03nk(nk+1)℘(z+ [Formula presented] |τ)+B]y(z),nk∈Z≥0 from the monodromy aspect. We prove the existence of a pre-modular form Zr,s n(τ) of weight [Formula presented] ∑nk(nk+1) such that the monodromy data (r,s) is characterized by Zr,s n(τ)=0. This generalizes the result in [17], where the Lamé case (i.e. n1=n2=n3=0) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations Δu+eu=16πδ0andΔu+eu=8π∑k=13δ [Formula presented] on a flat torus has the same number of even solutions. This result is quite surprising from the PDE point of view.

AB - In this paper, the second in a series, we continue to study the generalized Lamé equation with the Treibich-Verdier potential y″(z)=[∑k=03nk(nk+1)℘(z+ [Formula presented] |τ)+B]y(z),nk∈Z≥0 from the monodromy aspect. We prove the existence of a pre-modular form Zr,s n(τ) of weight [Formula presented] ∑nk(nk+1) such that the monodromy data (r,s) is characterized by Zr,s n(τ)=0. This generalizes the result in [17], where the Lamé case (i.e. n1=n2=n3=0) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations Δu+eu=16πδ0andΔu+eu=8π∑k=13δ [Formula presented] on a flat torus has the same number of even solutions. This result is quite surprising from the PDE point of view.

KW - Generalized Lamé equation

KW - Mean field equation

KW - Pre-modular form

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U2 - 10.1016/j.matpur.2019.05.004

DO - 10.1016/j.matpur.2019.05.004

M3 - Article

AN - SCOPUS:85066922600

SN - 0021-7824

VL - 132

SP - 251

EP - 272

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

ER -