走向现代数学学术报告 - 帅伟副教授（No. 935)

报告题目：Existence of positive solution for Schr\"odinger-Newton system with a doping profile

报告时间：2026年5月16日 14:30

报告地点：东海岸校区-D实315

报 告 人：帅伟 教授（华中师范大学）

邀 请 人：王大斌教授、谢华飞副教授

报告摘要：In this talk, we study the following Schr\"odinger-Newton system

\begin{equation}\label{0.1}

\left\{

\renewcommand{\arraystretch}{1.25}

\begin{array}{ll}

-\Delta u+u=\Phi u, \ & x\in\R^3,\\

-\Delta \Phi =u^2+\rho(x), \ & x\in\R^3,

\end{array}

\right.

\end{equation}

where $\rho \in L^{\frac{6}{5}}(\mathbb{R}^3)$ is a doping profile function. The existence and nonexistence of a ground state solution is established by using variational methods. Interestingly, the existence of solutions is significantly affected by the sign of $\rho(x)$. Specifically, we demonstrate that system \eqref{0.1} possesses a positive ground state solution if $\rho(x)\geq 0$ and $\|\rho\|_{L^{\frac{6}{5}}(\mathbb{R}^3)}$ is appropriately small. Conversely, system \eqref{0.1} does not have a ground state solution if $\rho(x) \leq 0$. However, by employing a global compactness lemma and a general minimax principle, we are succeed in showing that a high energy positive solution exists for system \eqref{0.1} if $\rho(x) \leq 0$ and $\|\rho\|_{L^{\frac{6}{5}}(\mathbb{R}^3)}$ is suitably small.

报告人简介：帅伟，理学博士，华中师范大学副教授，博士生导师。2016年博士毕业于华中师范大学，师从邓引斌教授。2016.12-2018.12 香港中文大学数学科学研究所助理研究员，合作导师为辛周平教授。主要研究方向是非线性椭圆形偏微分方程、非线性泛函分析。主要成果发表在J. Funct. Anal., Calc. Var. Partial Differential Equations, J. Differential Equations，Nonlinearity等国际期刊上。主持完成国家自然科学基金青年和面上等项目，现主持国家自然科学基金面上项目1项。