Electron. J. Diff. Equ., Vol. 2012 (2012), No. 85, pp. 1-36.

Existence and concentration of semiclassical states for nonlinear Schrodinger equations

Shaowei Chen

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.

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Shaowei Chen
School of Mathematical Sciences
Capital Normal University
Beijing 100048, China
email: chensw@amss.ac.cn

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