Morse Index of Multiple Blow-up Solutions to the Two-Dimensional Sinh-Poisson Equation (Analysis in Theory and Applications, 2022)
In this paper we consider the Dirichlet problemwhere $ρ$ is a small parameter and $Ω$ is a $C^2$ bounded domain in $\mathbb{R}^2$. In [1], the author 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.
Keywords: Morse index; sinh-Poisson equation; eigenvalues estimates.