# PDE Seminar

## Least energy solutions of fractional Schr\"odinger equations involving potential wells

### 唐仲伟 教授 (北京师范大学)

#### 2016年12月2日14:00-15:00 闵行校区行政楼12楼PDE中心1202报告厅

Abstract. In this talk, we study a class of nonlinear Schr\"odinger equations involving the fractional Laplacian. We assume that the potential of the equations includes a parameter $\lambda$. Moreover, the potential behaves like a potential well when the parameter $\lambda$ is large. Using variational methods, combining Nehari methods, we prove that the equation has a least energy solution which, as the parameter $\lambda$ large, localizes near the bottom of the potential well. Moreover, if the zero set int $V^{-1}(0)$ of $V(x)$ includes more than one isolated component, then $u_\lambda(x)$ will be trapped around all the isolated components. However, in Laplacian case when $s=1$, for $\lambda$ large, the corresponding least energy solution will be trapped around only one isolated component and will become arbitrary small in other components of int $V^{-1}(0)$. This is the essential difference with the Laplacian problems since the operator $(-\Delta)^{s}$ is nonlocal.