Multiple Semiclassical Standing Waves for Fractional Nonlinear Schr\"{o}dinger Equations

TL;DR

This paper constructs multiple semiclassical solutions for fractional nonlinear Schrödinger equations using Lyapunov-Schmidt reduction, revealing the existence of solutions related to the topology of the potential's critical manifold.

Contribution

It introduces a novel application of Lyapunov-Schmidt reduction to fractional Schrödinger equations, establishing the existence of multiple solutions tied to the critical manifold's topology.

Findings

Existence of multiple solutions when \\varepsilon is small

Solutions correspond to the cup length of the critical manifold

Applicable for a range of nonlinear exponents p

Abstract

Via a Lyapunov-Schmidt reduction, we obtain multiple semiclassical solutions to a class of fractional nonlinear Schr\"odinger equations. Precisely, we consider \begin{equation*} \varepsilon^{2s}(-\Delta)^{s}u+u+V(x)u=|u|^{p-1}u,\quad u\in H^s(\mathbf R^n), \end{equation*} where $0<s<1$, $n>4-4s$, $1<p<\frac{n+2s}{n-2s}$ (if $n>2s$) and $1<p<\infty$ (if $n\le 2s$), $V(x)$ is a non-negative potential function. If $V$ is a sufficiently smooth bounded function with a non-degenerate compact critical manifold $M$, then, when $\varepsilon$ is sufficiently small, there exist at least $l(M)$ semiclassical solutions, where $l(M)$ is the cup length of $M$.