Morse index, topological degree and local uniqueness of multi-spike solutions

We consider multi-spike positive solutions to the Lane-Emden problem in a bounded smooth planar domain, and investigate the related linearized eigenvalue problem. By obtaining estimates for its eigenvalues and sharp descriptions of the asymptotic behavior of the corresponding eigenfunctions, we compute the Morse index of each \(k\)-spike solution and the total degree of all the \(k\)-spike solutions concentrating at a non-degenerate critical points of the K-R function. As a consequence, we prove the local uniqueness of such solutions.