Normalized bound state solutions for the fractional Schr\"{o}dinger equation with potential

TL;DR

This paper establishes the existence of normalized bound state solutions for a fractional Schrödinger equation with a potential, using variational methods under various conditions on the potential function.

Contribution

It provides new existence results for normalized solutions of fractional Schrödinger equations with potential functions, extending previous work to broader conditions.

Findings

Existence of solutions proven via minimax principle.

Solutions are bounded and normalized with prescribed mass.

Results applicable to a range of potential functions.

Abstract

In this paper, we study the following fractional Schr\"{o}dinger equation with prescribed mass \begin{equation*} \left\{ \begin{aligned} &(-\Delta)^{s}u=\lambda u+a(x)|u|^{p-2}u,\quad\text{in $\mathbb{R}^{N}$},\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},\quad u\in H^{s}(\mathbb{R}^{N}), \end{aligned} \right. \end{equation*} where $0<s<1$, $N>2s$, $2+\frac{4s}{N}<p<2_{s}^{*}:=\frac{2N}{N-2s}$, $c>0$, $\lambda\in \mathbb{R}$ and $a(x)\in C^{1}(\mathbb{R}^{N},\mathbb{R}^{+})$ is a potential function. By using a minimax principle, we prove the existence of bounded state normalized solution under various conditions on $a(x)$.