Topics in higher order elliptic equations in the whole space

Date
2020/01/17 Fri 15:30 - 17:30
Room
3号館251号室
Speaker
Ngô Quốc Anh
Affiliation
Vietnam National University, Hanoi / The University of Tokyo
Abstract

Higher order elliptic equations with power-type nonlinearity of the form  $$ \Delta^m u = \pm u^\alpha $$ in an unbounded subset of $\mathbb R^n$ with $n \geq 1$, $m \geq 1$, and $\alpha \in \mathbb R$ often appear in many research works especially when $m \leq 2$ because they can be used to model many phenomena in reality and applied science. From the geometric point of view, the motivation and importance of studying solutions to these equations stem from the understanding of conformally flat manifolds as well as the well-known Yamabe problem, which can be seen from the seminal work of R. Schoen and S.T. Yau in 1988. Higher-order curvatures such as Q-curvature also constitute the above equations. Unfortunately, it becomes evident that, after putting together all the known results of the scientific literature, the knowledge on this class of equations still appears quite fragmentary even for basic information such as existence and non-existence. In this talk, first I will present a complete picture of the existence and non-existence of classical positive/non-negative solutions to the above equations in the full range of the parameters $n$, $m$, and $\alpha$. Then, in the existing regime, I will show whenever a maximum principle result is available. Next, I will focus on the fourth-order equations, namely $m=2$, and provide a complete classification of radial solutions in terms of asymptotic behavior. Then I will show how to use this classification to construct non-radial solutions. Geometric interpretation of these findings, as well as interesting questions, is also mentioned.