论文标题
二维SINH-POISSON方程的多重爆破解决方案的摩尔斯索引
Morse Index of Multiple Blow-Up Solutions to the Two-Dimensional Sinh-Poisson Equation
论文作者
论文摘要
在本文中，我们考虑了dirichlet问题\ begin {equation} \ label {iniz} \ begin {cases}-Δu=ρ^2（e^{u} - e^{ - e^{ - u}）＆\ text {in}是一个小参数，$ω$是$ \ mathbb {r}^2 $中的$ c^2 $界域。 [1]证明存在$ M $ - 点爆破解决方案$U_ρ$与其渐近行为共同存在。我们根据此问题相关的汉密尔顿功能的Morse索引计算$U_ρ$的Morse索引。此外，我们对第一个400万美元的特征值和特征功能进行了渐近估计。
In this paper we consider the Dirichlet problem \begin{equation} \label{iniz} \begin{cases} -Δu = ρ^2 (e^{u} - e^{-u}) & \text{ in } Ω\\ u=0 & \text{ on } \partial Ω, \end{cases} \end{equation} where $ρ$ is a small parameter and $Ω$ is a $C^2$ bounded domain in $\mathbb{R}^2$. [1] proves the existence of a $m$-point blow-up solution $u_ρ$ jointly with its asymptotic behaviour. we compute the Morse index of $u_ρ$ in terms of the Morse index of the associated Hamilton function of this problem. In addition, we give an asymptotic estimate for the first $4m$ eigenvalues and eigenfunctions.
