Non-degeneracy and local uniqueness of positive solutions to the Lane-Emden problem in dimension two

We are concerned with the Lane-Emden problem −∆u = u^P in Ω u > 0 in Ω u = 0 on ∂Ω, where Ω ⊂ R^2 is a smooth bounded domain and p > 1 is sufficiently large. Improving some known asymptotic estimates on the solutions, we prove the non-degeneracy and local uniqueness of the multi-spikes positive solutions for general domains. Our methods mainly use ODE’s theory, various local Pohozaev identities, blow-up analysis and the properties of Green’s function