论文标题
集中现象到更高阶段的liouville方程式
Concentration phenomena to a higher order Liouville equation
论文作者
论文摘要
我们研究了一系列解决方案$（u_k）$的爆炸和量化现象\int_Ωe^{2nu_k} dx \ leq c，$$在$ q_k $上的自然假设下。众所周知，直到一个子序列，要么$（u_k）$具有适当的规范，要么存在$β_k\ to \ infty $，以至于$ u_k =β_k（φ+o（1）$ in $ω\ setMinus in $ setminus（s_1 \ cups_φs_φ）$ nontrivial $ nontrivial nontrivial nontrivial nontrivial and $ nontrivim nontrivim nontrivim notivem- $ s_1 $，其中$ sφ$是$φ$的零集。我们证明了总曲率$ \ int _ {\tildeΩ} q_ke^{2nu_k} dx $上$ \tildeΩ\ subset（ω\ setMinuss_φ）$上的q_ke^{2nu_k} dx $的量化。我们还考虑了第三维中的非本地情况。
We study blow-up and quantization phenomena for a sequence of solutions $(u_k)$ to the prescribed $Q$-curvature problem $$ (-Δ)^nu_k= Q_ke^{2nu_k}\quad \text{in }Ω\subset\mathbb{R}^{2n},\quad \int_Ωe^{2nu_k}dx\leq C,$$ under natural assumptions on $Q_k$. It is well-known that, up to a subsequence, either $(u_k)$ is bounded in a suitable norm, or there exists $β_k\to\infty$ such that $ u_k=β_k(φ+o(1))$ in $Ω\setminus (S_1\cup S_φ)$ for some non-trivial non-positive $n$-harmonic function $φ$ and for a finite set $S_1$, where $S_φ$ is the zero set of $φ$. We prove quantization of the total curvature $\int_{\tildeΩ}Q_ke^{2nu_k}dx$ on the region $\tildeΩ\Subset(Ω\setminus S_φ)$. We also consider a non-local case in dimension three.
