TY - JOUR

T1 - EVEN SOLUTIONS OF SOME MEAN FIELD EQUATIONS AT NON-CRITICAL PARAMETERS ON A FLAT TORUS

AU - Kuo, Ting Jung

AU - Lin, Chang Shou

N1 - Funding Information:
Received by the editors January 31, 2021, and, in revised form, June 8, 2021, June 14, 2021, June 15, 2021, and June 16, 2021. 2020 Mathematics Subject Classification. Primary 35J15, 35J60, 34M45, 34M03. The research of the first author was supported by MOST 107-2628-M-003-002-MY4.
Publisher Copyright:
© 2022 American Mathematical Society

PY - 2022

Y1 - 2022

N2 - In this paper, we consider the mean field equation Δu + eu = Σ3i=0 4πniδωi/2 in Eτ , where ni ∈ ℤ≥0, Eτ is the flat torus with periods ω1 = 1, ω2 = τ and Im τ > 0. Assuming N = Σ3i=0 ni is odd, a non-critical case for the above PDE, we prove: (i) If among {ni|i = 0, 1, 2, 3} there is only one odd integer, then there is always an even solution. Furthermore, if n0 = 0 and n3 is odd, then up to SL2(Z) action, except for finitely many Eτ , there are exactly n3+1/2 even solutions. (ii) If there are exactly three odd integers in {ni|i = 0, 1, 2, 3}, then the equation has no even solutions for any flat torus Eτ . Our second result might suggest the symmetric solution of the above mean field equation does not hold in general.

AB - In this paper, we consider the mean field equation Δu + eu = Σ3i=0 4πniδωi/2 in Eτ , where ni ∈ ℤ≥0, Eτ is the flat torus with periods ω1 = 1, ω2 = τ and Im τ > 0. Assuming N = Σ3i=0 ni is odd, a non-critical case for the above PDE, we prove: (i) If among {ni|i = 0, 1, 2, 3} there is only one odd integer, then there is always an even solution. Furthermore, if n0 = 0 and n3 is odd, then up to SL2(Z) action, except for finitely many Eτ , there are exactly n3+1/2 even solutions. (ii) If there are exactly three odd integers in {ni|i = 0, 1, 2, 3}, then the equation has no even solutions for any flat torus Eτ . Our second result might suggest the symmetric solution of the above mean field equation does not hold in general.

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U2 - 10.1090/proc/15721

DO - 10.1090/proc/15721

M3 - Article

AN - SCOPUS:85124597748

SN - 0002-9939

VL - 150

SP - 1577

EP - 1590

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 4

ER -