Morse index, topological degree and local uniqueness of multi-spikes solutions to the Lane-Emden problem in dimension two

We consider multi-spike positive solutions to the Lane-Emden problem in any bounded smooth planar domain and compute their Morse index, extending to the dimension $N=2$ classical theorems due to Bahri-Li-Rey (1995) and Rey (1999) when $N\ge 4$ and $N=3$, respectively. Furthermore, by deeply investigating their concentration behavior, we also derive the total topological degree. The Morse index and the degree counting formula yield a new local uniqueness result.