Concentrating standing waves for the fractional nonlinear Schrödinger equation

Abstract

We consider the semilinear equation $${\epsilon }^{2s}(-\Delta {)}^{s}u+V(x)u-u^p=0,u>0,u\in H^{2s}({\mathbb{R}}^N)$$ where $0<s<1,1<p<\frac{N+2s}{N-2s}$, $V(x)$ is a sufficiently smooth potential with ${\inf }_{\mathbb{R}}V(x)>0$, and $\epsilon >0$ is a small number. Letting $w_{\lambda }$ be the radial ground state of $(-\Delta {)}^s{w}_{\lambda }+\lambda w_{\lambda }-w_{\lambda }^p=0$ in $H^{2s}({\mathbb{R}}^N)$, we build solutions of the form $$u_{\epsilon }(x)\sim \sum _{i=1}^k{w}_{{\lambda }_i}((x-{\xi }_i^{\epsilon })/\epsilon ),$$ where ${\lambda }_i=V({\xi }_i^{\epsilon })$ and the ${\xi }_i^{\epsilon }$ approach suitable critical points of $V$. Via a Lyapunov Schmidt variational reduction, we recover various existence results already known for the case $s=1$. In particular such a solution exists around $k$ nondegenerate critical points of $V$. For $s=1$ this corresponds to the classical results by Floer-Weinstein and Oh.

26 pages

References in corpus (1)

• Concentration phenomenon for fractional nonlinear Schrödinger equations