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.