# Postdoc Seminar

## A remark on Lane-Emden equation and the corresponding Dirichlet problem

### 李芳 博士 (华师大PDE中心博士后)

#### 2012年11月20日(周二)下午 1:00-2:50 闵行校区行政楼12楼PDE中心1202报告厅

Abstract. It is known that for the equation $\Delta u+|u|^{p-1}u=0$ over the whole space $\mathbb R^n$, where $n>2$ and $p>(n+2)/(n-2)$, all regular radial solutions oscillate around an explicit singular radial solution when $p < p_c$, while no such oscillations occur in the remaining case $p\geq p_c$. Moreover, for the Dirichlet problem for the equation $\Delta u+\lambda(1+u)^p=0$ over the unit ball in $\mathbb R^n$, where $\lambda > 0$ is a parameter, it is known that the extremal solution is regular as long as $p < p_c$ and singular if $p \geq p_c$. In this talk, we will present a simple explanation about why the same critical exponent $p_c$ appears in the above two equations.