Invited by Prof. Lu Yang and Prof. Jun Geng of School of Mathematics and Statistics, Prof. Wei Long from School of Mathematics and Statistics，Jiangxi Normal Universitywill give a lecture.

Title: Henon equation in a general domain by Pohozaev identities and Reduction Method

Time: 10:00 a.m.,Jul. 20th, 2021

Location: Room 518, Polytechnic building

Abstract: In this talk, we are concerned with the following H\'{e}non problem \begin{equation*}\label{eq1} \left\{\begin{array}{ll} \Delta u= |x|^\alpha u^{2^*-1-\varepsilon} ,\ \ u>0,\ \hspace{3mm} & \text{ in } \ \ \Omega,\\

u=0,\ \ \hspace{3mm}&\text{ on } \ \ \partial\Omega，\end{array} \right.

\end{equation*} where $N\geq 4$, $2^*=\frac{2N}{N-2}$, $\alpha >0$, $\ep$ is a small positive parameter, $\Omega $ is a smooth bounded domain in $\r^N$ and $0 \in \Omega$. Most of previous works for H\'{e}non problems were investigated in special domains, such as balls and annulus. In this paper, we will study the case when $\Omega$ is a more general domain, which does not satisfy symmetry any more. We first investigate the necessary condition on the location of the blow up point for the peak solution to the above H\'{e}non problem. Then, we prove that, as $\varepsilon\rightarrow 0$, the above problem has a positive solution with multiple bubbles under a suitable condition on the geometry of $\Omega$.