Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 85, pp. 1-36.

Title: Existence and concentration of semiclassical states
       for nonlinear Schrodinger equations

Author: Shaowei Chen (Capital Normal Univ., Beijing, China)

Abstract:
 In this article, we study the semilinear Schrodinger equation
 $$
 -\epsilon^2\Delta u+ u+ V(x)u=f(u),\quad u\in H^1(\mathbb{R}^N),
 $$
 where $N\geq 2$ and $\epsilon>0$ is a small parameter.
 The function $V$  is bounded in $\mathbb{R}^N$,
 $\inf_{\mathbb{R}^N}(1+V(x))>0$ and it has a possibly degenerate
 isolated critical point.  Under some conditions on f, we prove
 that as $\epsilon\to 0$, this equation has a solution which concentrates
 at the  critical point of V.

Submitted August 16, 2011. Published May 31, 2012.
Math Subject Classifications: 35J20, 35J70.
Key Words: Semilinear Schrodinger equation; variational reduction method.