报告题目: Sign-changing solutions for critical Hamiltonian systems

报告人: 严树森教授（华中师范大学）

报告时间: 2026年01月06日08:30-10:30

报告地点: 数学与统计学院135报告厅

报告摘要: We build infinitely many geometrically distinct non-radial sign-changing solutions for the Hamiltonian-type elliptic systems

$$ -\Delta u =|v|^{p-1}v\ \hbox{in}\ \mathbb R^N,\

-\Delta v =|u|^{q-1}u\ \hbox{in}\ \mathbb R^N,$$

where the exponents $(p,q)$ satisfy $p,q>1$ and belong to the critical hyperbola

$$\frac1{p+1}+\frac1{q+1} =\frac {N-2}N.$$

To establish this result, we introduce several new ideas and strategies that are both robust and potentially applicable to other critical problems lacking the Kelvin invariance.

报告人简介: 严树森：华中师范大学教授，研究方向为非线性椭圆方程，在关于奇异椭圆问题的多峰解的存在性，Ambrosetti-Prodi型椭圆方程解的个数的估计的Lazer-McKenna猜想，非紧椭圆问题无穷多个正解的存在性，Chern-Simons方程和二维不可压流体的涡补丁问题爆破解与Kirchhoff-Routh函数之间的关系，椭圆问题的波峰解的非退化性与局部唯一性等做出了重要的工作，得到了国际同行的高度认可。

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

报告人: 郭玉霞教授（清华大学数学系）

报告时间: 2026年01月06日08:30-10:30

报告地点: 数学与统计学院135报告厅

报告摘要: 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.

报告人简介: 郭玉霞：清华大学数学系教授，博士生导师，德国洪堡基金获得者。主要从事非线性泛函分析及其在偏微分方程中的应用等方面的研究工作。2002年世界数学家大会卫星会议邀请报告人。2002年以来曾先后主持完成国家自然科学基金6项，作为主要成员参与完成重点项目1项.目前参重点项目1项, 主持面上项目1项。公开发表国际SCI论文100余篇,部分研究成果发表在国际权威数学刊比如：Comm.Pure. Appl. Math., Math Anal.Jour. Diff. Equa., Comm.Parl. Diff. Equa., Cal. Var. PDE., Jour. Func. Anal.， SIAMJ. Contr. Opt., Pro. Lond. Math Soc, JMPA等，其研究成果被专家学者广泛引用。