论文标题
Moser-trudinger的临界点在闭合表面上起作用
Critical points of the Moser-Trudinger functional on closed surfaces
论文作者
论文摘要
鉴于闭合的Riemann表面$（σ，G）$和任何正平滑重量，我们使用Minmax方案以及紧凑的量化，量化结果以及尖锐的能量估计来证明功能的正临界点存在$ j_ {p，β}（u）= \ frac {2-p} {2} {2} \ left（\ frac {p \ | u \ | _ {h^1}^2} {2β} {2β} \ right） f dv_g，（1,2）$和$β> 0 $，{或}的每$ p \ in（0，\ infty）\ setminus4π\ mathbb {n} $。让$ p \ uparrow 2 $我们获得Moser-trudinger功能$$ f（u）的积极关键点：= \int_σ（e^{u^2} -1）f dv_g $$约束至$ \ mathcal {e}_β： } \ | v \ | _ {h^1}^2 =β\ right \} $对于任何$β> 0 $。
Given a closed Riemann surface $(Σ,g)$ and any positive smooth weight, we use a minmax scheme together with compactness, quantization results and with sharp energy estimates to prove the existence of positive critical points of the functional $$J_{p,β}(u)=\frac{2-p}{2}\left(\frac{p\|u\|_{H^1}^2}{2β} \right)^{\frac{p}{2-p}}-\ln \int_Σ(e^{u_+^p}-1) f dv_g,$$ for every $p\in (1,2)$ and $β>0$, {or} for $p=1$ and $β\in (0,\infty)\setminus 4π\mathbb{N}$. Letting $p\uparrow 2$ we obtain positive critical points of the Moser-Trudinger functional $$F(u):=\int_Σ(e^{u^2}-1)f dv_g$$ constrained to $\mathcal{E}_β:=\left\{v\text{ s.t. }\|v\|_{H^1}^2=β\right\}$ for any $β>0$.
