91论坛 数学大讲坛第七期

第六十七讲——清华大学郭玉霞教授学术报告

题目： Multiple Boundary Peak Solution for Critical Hamiltonian System with Neumann boundary

时间：2025年12月25日(星期四) 上午10:30-12:00

地点：腾讯会议（会议ID：956-519-889）

报告人：郭玉霞 教授

摘要：We consider the following elliptic system with Neumann boundary:

\begin{equation*}

\begin{cases}

-\Delta u + \mu u=v^p,\;\;\; &\hbox{in } \Omega,\\

-\Delta v + \mu v=u^q,\;\;\; &\hbox{in } \Omega,\\

\frac{\partial u}{\partial n} = \frac{\partial v}{\partial n} = 0, &\hbox{on } \partial\Omega,\\

u>0,v>0, &\hbox{in } \Omega,

\end{cases}

\end{equation*}

where $\Omega \subset \R^N$ is a smooth bounded domain, $\mu$ is a positive constant and $(p,q)$ lies in the critical hyperbola:

$$

\dfrac{1}{p+1} + \dfrac{1}{q+1} =\dfrac{N-2}{N}.

$$

By using the Lyapunov-Schmidt reduction technique, we establish the existence of infinitely many solutions to above system. These solutions have multiple peaks that are located on the boundary $\partial \Omega$. Our results show that the geometry of the boundary $\partial\Omega,$ especially its mean curvature, plays a crucial role on the existence and the behaviour of the solutions to the problem.