Multiple Semiclassical Standing Waves for Fractional Nonlinear Schrödinger Equations

Abstract

Via a Lyapunov-Schmidt reduction, we obtain multiple semiclassical solutions to a class of fractional nonlinear Schrödinger equations. Precisely, we consider \begin{equation*} \varepsilon^{2s}(-Δ)^{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 $\epsilon$ is sufficiently small, there exist at least $l(M)$ semiclassical solutions, where $l(M)$ is the cup length of $M$.

Some error corrected, several references added

References in corpus (2)

• Concentration phenomenon for fractional nonlinear Schrödinger equations
• Uniqueness and nondegeneracy of positive solutions of $\Ds u+u=u^p$ in ${\mathbb{R}}^N$ when $s$ is close to 1

Cited by in corpus (5)

• On the fractional NLS equation and the effects of the potential well's topology
• Multiplicity and concentration results for local and fractional NLS equations with critical growth
• Nondegeneracy of ground states and multiple semiclassical solutions of the Hartree equation for general dimensions
• Singularly perturbed Neumann problem for fractional Schrödinger equations
• Higher topological type semiclassical states for fractional nonlinear elliptic equations