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

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 $a(x)$.