Local uniqueness of concentrated solutions and some applications on nonlinear Schr?dinger equations with very degenerate potentials

摘要

We revisit the following nonlinear Schrodinger equation-epsilon 2 Delta u + V(x)u = up-1, u > 0, u is an element of H1(RN), where epsilon > 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).

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