Normalized bound state solutions of fractional Schrödinger equations with general potential

Abstract

In this paper, we study a class of fractional Schrödinger equation \begin{equation} \label{eq0} \left\{ \begin{aligned} &(-Δ)^{s}u=λu+a(x)|u|^{p-2}u,\\ &\int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},\ u\in H^{s}(\mathbb{R}^{N}), \end{aligned} \right. \end{equation} where $N>2s$, $s\in (0,1)$ and $p\in (2,2+4s/N),c>0$. $a(x)\in C({\mathbb{R}}^N,\mathbb{R})$ is a positive potential function. By using Fixed Point Theorem of Brouwer, barycenter function and variational method, we obtain the existence of normalized bound solutions for the problem.