摘要： This talk is related to a singularly perturbed nonlinear Schrödinger equation. We obtain a more accurate location for the concentrated points, the existence and the local uniqueness for positive peak solutions when the potential V(x) possesses non-isolated critical points by using the modified finite dimensional reduction method based on local Pohozaev identities. Moreover, for several special potentials, with its critical point set being a lower dimensional ellipsoid, or a part of hyperboloid of one sheet or two sheets, we obtain the number and symmetry of k-peak solutions by using local uniqueness of concentrated solutions. Here the main difficulty comes from the different degenerate rate along different directions at the critical points of V(x).