Multiplicity of normalized solutions for the fractional Schrödinger equation with potentials

Abstract

We get multiplicity of normalized solutions for the fractional Schrödinger equation $$ (-Δ)^su+V(\varepsilon x)u=λu+h(\varepsilon x)f(u)\quad \mbox{in $\mathbb{R}^N$}, \qquad\int_{\mathbb{R}^N}|u|^2dx=a, $$ where $(-\Delta {)}^s$ is the fractional Laplacian, $s\in (0,1)$, $a,\epsilon >0$, $\lambda \in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $V,h:{\mathbb{R}}^N\to [0,+\infty )$ are bounded and continuous, and $f$ is continuous function with $L^2$-subcritical growth. We prove that the numbers of normalized solutions are at least the numbers of global maximum points of $h$ when $\epsilon$ is small enough.

arXiv comment19 pages, revised version