Multiplicity of normalized solutions for the fractional Schr\"{o}dinger equation with potentials

TL;DR

This paper proves the existence of multiple normalized solutions for a fractional Schrödinger equation with potentials, showing the number of solutions relates to the number of global maxima of a potential function when a parameter is small.

Contribution

It establishes the multiplicity of solutions for the fractional Schrödinger equation with potentials, linking the number of solutions to the critical points of a potential function.

Findings

Number of solutions at least equals the number of global maxima of h

Solutions exist for small enough ε

Solutions are normalized with fixed L^2 norm

Abstract

We get multiplicity of normalized solutions for the fractional Schr\"{o}dinger equation $$ (-\Delta)^su+V(\varepsilon x)u=\lambda u+h(\varepsilon x)f(u)\quad \mbox{in $\mathbb{R}^N$}, \qquad\int_{\mathbb{R}^N}|u|^2dx=a, $$ where $(-\Delta)^s$ is the fractional Laplacian, $s\in(0,1)$, $a,\varepsilon>0$, $\lambda\in\mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $V,h:\mathbb{R}^N\rightarrow[0,+\infty)$ 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 $\varepsilon$ is small enough.