具有临界指数非线性的2 m阶椭圆型方程的量子化
Quantization for an elliptic equation of order 2m with critical exponential non-linearity
摘要：
在光滑有界区域$Ω\subset\R{2m}$上，考虑方程$(-Δ)^m u_k=λ_k u_k e^{mu_k^2}$在Dirichlet边界条件下的正解序列$u_k\stackrel{w}{\rightharpoondown} 0$在$H^m(Ω)$中的存在性，其中$0<λ_k\to 0$ .假设$$Λ:=\lim_{k\to\infty}\int_Ωu_k(-Δ)^m u_k dx<\infty,$$，我们证明$Λ$是$Λ_1:=(2m-1)!\vol(S^{2m})$的整数倍，总$Q$ -曲率的标准$2m$维球体。
英文摘要：
On a smoothly bounded domain $Ω\subset\R{2m}$ we consider a sequence of positive solutions $u_k\stackrel{w}{\rightharpoondown} 0$ in $H^m(Ω)$ to the equation $(-Δ)^m u_k=λ_k u_k e^{mu_k^2}$ subject to Dirichlet boundary conditions, where $0<λ_k\to 0$. Assuming that $$Λ:=\lim_{k\to\infty}\int_Ωu_k(-Δ)^m u_k dx<\infty,$$ we prove that $Λ$ is an integer multiple of $Λ_1:=(2m-1)!\vol(S^{2m})$, the total $Q$-curvature of the standard $2m$-dimensional sphere.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
