We revisit the following nonlinear Schrdinger equation where ɛ > 0 is a small parameter, N ⩾ 2, 2 < p < 2*. Here we are mainly concerned with local uniqueness and the number of concentrated solutions of the above nonlinear Schrdinger equation for a kind of non-admissible potential V(x) which possesses non-isolated critical points. First, we establish a more accurate location for the concentrated points, which will need an observation on the structure of the potential. Next, we prove local uniqueness for positive single-peak solutions. Then some results concerning on the number and symmetry of single-peak solutions are also given. To our knowledge, this seems to be the first result on local uniqueness and the number of concentrated solutions with non-admissible potential, which generalizes Grossi’s results (2002 Inst. H. Poincare Anal. Non Lineaire 19 261–80)

我们重新审视以下非线性 Schrdinger 方程，其中ɛ > 0 是一个小参数，N ⩾ 2, 2 < p < 2*。这里我们主要关心的是上述非线性薛定谔方程的局部唯一性和集中解的个数，对于一种不可容许势V ( x) 具有非孤立的临界点。首先，我们为集中点建立一个更准确的位置，这将需要对势的结构进行观察。接下来，我们证明正单峰解的局部唯一性。然后还给出了关于单峰解的数量和对称性的一些结果。据我们所知，这似乎是关于局部唯一性和具有不可容许潜力的集中解的数量的第一个结果，它概括了 Grossi 的结果 (2002 Inst. H. Poincare Anal. Non Lineaire 19 261–80)