Local uniqueness of concentrated solutions and some applications on nonlinear Schrödinger equations with very degenerate potentials

Abstract

We revisit the following nonlinear Schrödinger equation−ε2Δu+V(x)u=up−1,u>0,u∈H1(RN), where ε>0 is a small parameter, N≥2 and 2<p<2⁎. We obtain a more accurate location for the concentrated points, the existence and the local uniqueness for positive k-peak solutions when 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 low-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).