Local uniqueness and non-degeneracy of bubbling solution for critical Hamiltonian system

In this paper, we consider the following elliptic system of Hamiltonian type on a bounded domain: {−Δu=K₁(|y|)|v|^p−1v, in B₁(0),−Δv=K₂(|y|)|u|^q−1u, in B₁(0),u=v=0 on ∂B₁(0), where K₁(r) and K₂(r) are positive bounded functions, B₁(0) is the unit ball in R^N, (p,q) is a pair of positive numbers lying on the critical hyperbola: [Formula presented] We investigate the local uniqueness and the nondegeneracy of the bubbling solution for problem (0.1). Our proof is based on the local Pohozaev identities, blow-up analysis, and the properties of Greens function. Our results are quite different from single equation, which is mainly caused by the non-cooperative nonlinear terms and (p,q) lying on the critical hyperbola. We believe that the various new ideas and technique computations that we used in this paper would be very useful to deal with other related problems of Hamiltonian type with interact critical exponents.