Concentration phenomenon for fractional nonlinear Schrödinger equations

Guoyuan Chen; Youquan Zheng

doi:10.3934/cpaa.2014.13.2359

Article Contents

2014, Volume 13, Issue 6: 2359-2376. Doi: 10.3934/cpaa.2014.13.2359

This issue Previous Article Finite speed of propagation for mixed problems in the $WR$ class Next Article On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping

Concentration phenomenon for fractional nonlinear Schrödinger equations

Guoyuan Chen^1, and
Youquan Zheng^2,

1.
School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou 310018, Zhejiang
2.
School of Science, Tianjin University, Tianjin 300072

Received: September 30, 2013

Revised: March 31, 2014

Published: October 2014

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Abstract

We study the concentration phenomenon for solutions of the fractional nonlinear Schrödinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation \begin{eqnarray} (-\varepsilon^2\Delta)^sv+Vv-|v|^{\alpha}v=0\quad\mbox{in}\quad\mathbf R^n, \end{eqnarray} where $n =1, 2, 3$, $\max\{\frac{1}{2}, \frac{n}{4}\}< s < 1$, $1 \leq \alpha < \alpha_*(s,n)$, $V\in C^3_{b}(\mathbf{R}^n)$. Here the exponent $\alpha_*(s,n)=\frac{4s}{n-2s}$ for $0 < s < \frac{n}{2}$ and $\alpha_*(s,n)=\infty$ for $s \geq\frac{n}{2}$. Then for each non-degenerate critical point $z_0$ of $V$, there is a nontrivial solution of equation (1) concentrating to $z_0$ as $\varepsilon\to 0$.

Keywords:

Fractional nonlinear Schrödinger equation,
concentration solutions,
Lyapunov-Schmidt reduction.

Mathematics Subject Classification: Primary: 35J10, 35J30; Secondary: 35J35, 35J61, 35J91.

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