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 $\Omega$ 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.