Location and uniqueness of concentration solutions

This talk is concerned with the following nonlinear Schr\"odinger equation$$-\varepsilon^2\Delta u+ V(x)u=|u|^{p-2}u,\,\,\,u\in H^1(R^N),$$ where $\varepsilon>0$ is a small parameter, $N\geq 1$, $2<p<2^*$. For a class of  $V(x)$ which possesses non-isolated critical points, we obtain the necessary condition, existence and local uniqueness of the positive single peak solution with concentrating at this kind of points. Here the main difficulty is the degeneracy and inhomogeneity of $V(x)$ at the  concentrating point.