Normalized bound state solutions of fractional Schr\"{o}dinger equations with general potential

TL;DR

This paper establishes the existence of normalized bound state solutions for a class of fractional Schrödinger equations with a general potential, using variational methods and fixed point theory.

Contribution

It introduces a novel approach combining Brouwer's Fixed Point Theorem and barycenter functions to prove existence of solutions for fractional Schrödinger equations with general potentials.

Findings

Existence of normalized bound solutions proven.

Application of variational methods to fractional equations.

Use of fixed point theorem in nonlinear PDE context.

Abstract

In this paper, we study a class of fractional Schr\"{o}dinger equation \begin{equation} \label{eq0} \left\{ \begin{aligned} &(-\Delta)^{s}u=\lambda u+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.