Morse Index of Multiple Blow-Up Solutions to the Two-Dimensional Sinh-Poisson Equation

In this paper we consider the Dirichlet problem \begin{equation} \label{iniz} \begin{cases} -\Delta u = \rho^2 (e^{u} - e^{-u}) & $\text{ in }$ \Omega\ u=0 & $\text{ on }$ \partial \Omega, \end{cases} \end{equation} where $\rho$ is a small parameter and $\mathrm{\Omega }$ is a $C^2$ bounded domain in ${\mathbb{R}}^2$. [1] proves the existence of a $m$-point blow-up solution $u_{\rho }$ jointly with its asymptotic behaviour. we compute the Morse index of $u_{\rho }$ 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.