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

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 $M$.