# PDE Seminar

## A complete description of the behavior of radial solutions to $\Delta^2 u = u^\alpha$ in $\mathbb{R}^n$ at infinity

### Quoc Anh Ngo (Vietnam National University at Hanoi and The University of Tokyo)

#### 2019年12月13日14:00-15:00 闵行校区数学楼102报告厅

Abstract: Higher-order elliptic PDEs often appear in many physical phenomena, in applied mathematics, or even in geometry. However, they often create difficulty when studying. The simplest model of such higher-order equations that one can consider is the bi-Laplace equations of the form $\Delta^2 u = ± u^\alpha$ with real $\alpha$ in the whole space $\mathbb{R}^n$. Exhaustive existence and non-existence results for these equations are now known. However, toward a complete understanding of the picture of solutions beyond the existence and non-existence results, it is necessary to look into the class of positive radial solutions to these equations in the existing regime. In this sense, it is tempting to understand the behavior of these solutions near infinity. While the behavior of radial solutions to $\Delta^2 u = - u^\alpha$ near infinity is mostly understood, much less is known for the case of $\Delta^2 u = u^\alpha$ where $\alpha$ is subcritical. In this talk, I describe a new method, which is surprisingly simple and easy to implement to other equations, to compute the exact growth at the infinity of radial solutions to $\Delta^2 u =u^\alpha$ in$\mathbb{R}^n$. This is joint work with V.H. Nguyen and Q.H. Phan.