Non-degeneracy of multi-peak solutions for the Schrodinger-Poisson problem

In this article, we consider the following Schrodinger-Poisson problem: {-epsilon(2)Delta u + V(y)u + Phi(y)u = vertical bar u vertical bar(p-1)u, y is an element of R-3, -Delta Phi(y) = u(2), y is an element of R-3, where epsilon > 0 is a small parameter, 1 < p < 5, and V(y) is a potential function. We construct multi-peak solution concentrating at the critical points of V(y) through the Lyapunov-Schmidt reduction method. Moreover, by using blow-up analysis and local Pohozaev identities, we prove that the multi-peak solution we construct is non-degenerate. To our knowledge, it seems be the first non-degeneracy result on the Schodinger-Poisson system.