Mean field equations and premodular forms, I: at critical parameter $16\pi$
Mean field equations and premodular forms, I: at critical parameter $16\pi$
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发表时间：
2016-10
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通讯作者：
Zhijie Chen;Ting-Jung Kuo;Changshou Lin
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作者：
Zhijie Chen;Ting-Jung Kuo;Changshou Lin
A conjecture about the mean field equation $\Delta u+e^{u}=8n\pi \delta_{0}$ on a flat torus $E_{\tau}$ is the non-existence of solutions if $\tau\in i\mathbb{R}^{+}$. For any $n\in \mathbb{N}_{\geq 2}$, this conjecture seems very challenging from the viewpoint of PDE theory. In order to solve this conjecture, a premodular form $Z_{r,s}^{(n)}(\tau)$ was introduced by Wang and the third author \cite{CLW2}, and is used to give necessary and sufficient conditions for the existence of solutions. In this paper, we succeed to prove the conjecture for $n=2$ (i.e. at critical parameter $16\pi$). In turn, we could apply this non-existence result to obtain the structure of zeros of the premodular form $Z_{r,s}^{(2)}% (\tau)$. As a consequence, we obtain the existence of solutions for $n=2$ if $\tau=\frac{1}{2}+ib$ and $b>b^{\ast}$ for some $b^{\ast}\in(\frac{\sqrt{3}}{2},\frac{6}{5})$.
A conjecture about the mean field equation $\Delta u+e^{u}=8n\pi \delta_{0}$ on a flat torus $E_{\tau}$ is the non-existence of solutions if $\tau\in i\mathbb{R}^{+}$. For any $n\in \mathbb{N}_{\geq 2}$, this conjecture seems very challenging from the viewpoint of PDE theory. In order to solve this conjecture, a premodular form $Z_{r,s}^{(n)}(\tau)$ was introduced by Wang and the third author \cite{CLW2}, and is used to give necessary and sufficient conditions for the existence of solutions. In this paper, we succeed to prove the conjecture for $n=2$ (i.e. at critical parameter $16\pi$). In turn, we could apply this non-existence result to obtain the structure of zeros of the premodular form $Z_{r,s}^{(2)}% (\tau)$. As a consequence, we obtain the existence of solutions for $n=2$ if $\tau=\frac{1}{2}+ib$ and $b>b^{\ast}$ for some $b^{\ast}\in(\frac{\sqrt{3}}{2},\frac{6}{5})$.