# Postdoc Seminar

## Existence and stability of entire solutions to the polyharmonic equation $(-\Delta )^m u=e^u$ in $\mathbb{R}^N$

### 黄侠 博士 (华师大PDE中心博士后)

#### 2016年12月20日(周二)下午 1:00-4:00 闵行校区行政楼12楼PDE中心1202报告厅

Abstract. We will report existence and stability properties of entire solutions of a polyharmonic equation with an exponential nonlinearity. As a first result on stability we will give that stable solutions (not necessarily radial) in dimensions lower than the conformal one never exist. On the other hand, in "supercritical dimensions" we will show the existence of many entire stable solutions. The existence of stable radial solutions on the borderline for $m \geq 4$ even in arbitrary supercritical dimension is a new phenomenon comparing to $m =2$, where the borderline solutions are not stable out of any compact set if $5 \leq N\leq 12$. This talk is based on the work of A. Farina - A. Ferrero[Ann. IHP. Anal, 2016], X. Huang - D. Ye[JDE, 2016].