arXiv subjects

Xin Bao

Publications and source records attributed to Xin Bao.

2 recordsLinked to original sources

arXiv2023

Normalized bound state solutions for the fractional Schrödinger equation with potential

In this paper, we study the following fractional Schrödinger equation with prescribed mass \begin{equation*} \left\{ \begin{aligned} &(-Δ)^{s}u=λu+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 $0 2s$, $2+\frac{4s}{N} 0$, $λ\in \mathbb{R}$ 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 $a(x)$.

math.AP↗

arXiv2023

Normalized bound state solutions of fractional Schrödinger equations with general potential

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.

math.AP↗